Df1> 


IN  MEMOR1AM 
FLOR1AN  CAJORI 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/elementaryalgebrOOIymarich 


ELEMENTARY  ALGEBRA 


BY 
ELMER  A.  LYMAN 

Professor  of  Mathematics  in  the  Michigan  State  Normal  College 
Ypsilanti,  Michigan 

AND 

ALBERTUS  DARNELL 

Head  of  Department  of  Mathematics,  Central  High  School 
Detroit,  Michigan 


AMERICAN  BOOK  COMPANY 

NEW  YORK  CINCINNATI  CHICAGO 


COPYRIGHT,    1917,   BY 

ELMEE  A.  LYMAN  and  ALBERTUS  DARNELL 

All  rights  reserved 

l.  and  d.  el.  alg. 

W.  P.    I 


PREFACE 

The  object  of  this  book  is  to  provide  a  complete  course  in 
Elementary  Algebra  that  will  satisfy  the  requirements  of 
courses  of  study  in  various  states  and  of  the  College  Entrance 
Board. 

Vitality  has  been  given  to  the  subject  by  imbuing  it  with 
the  interest  that  accrues  from  connection  with  problems  of 
everyday  life  and  by  careful  correlation  with  arithmetic.  The 
utility  of  algebra  is  emphasized  from  the  start  by  showing  how 
much  easier  it  is  to  solve  certain  problems  by  algebra  than  by 
arithmetic. 

Simplicity  is  the  keynote  of  the  book.  This  effect  is  gained 
by  omitting  exercises  of  undue  difficulty  as  well  as  trouble- 
some phases  of  the  subject  that  are  not  essential.  A  careful 
development  of  each  new  principle  anticipates  difficulties  ;  and 
abundant  illustrations  and  examples  give  further  emphasis  to 
the  point  at  issue. 

The  easy  oral  exercises  assist  in  developing  and  fixing  in 
mind  the  principles  and  processes.  The  written  exercises  are 
very  abundant  and  range  from  the  simplest  type  to  some  of 
sufficient  difficulty  to  test  the  pupil's  power  and  to  provide 
adequate  drill.  Many  of  the  exercises  are  taken  from  entrance 
examination  questions  set  by  various  colleges  and  universities, 
the  source  being  indicated  in  all  such  cases. 

The  problems  are  practical  and  varied.  They  include  appli- 
cations to  geometry,  physics,  engineering,  agriculture  and  com- 
merce, and  various  interests  of  everyday  life. 

iii 


M306039 


iv  Preface 

The  importance  of  the  equation  is  recognized  throughout  by- 
abundance  of  practice. 

Thoroughness  and  accuracy  are  secured,  first,  by  many  re- 
views, and  second,  by  the  emphasis  placed  on  the  checking  of 
results. 

Exercises  designed  to  encourage  an  intelligent  translation  of 
algebraic  language  are  provided  early  in  the  book.  The  appli- 
cation of  algebraic  principles  to  solutions  by  formulas  has  also 
been  emphasized. 

Graphical  representation  is  introduced  in  two  chapters,  but 
is  so  arranged  that  it  may  be  omitted  at  the  option  of  the 
teacher  without  interrupting  the  sequence  of  the  work. 

ELMER   A.   LYMAN. 
ALBERTUS  DARNELL. 


CONTENTS 


PAGB 

I.     Introduction    ...   * 1 

II.    Positive  and  Negative  Numbers 15 

Addition  of  Signed  Numbers 20 

Subtraction  of  Signed  Numbers 22 

Multiplication  of  Signed  Numbers    .        .        .        .        .25 

Division  of  Signed  Numbers 27 

III.  Addition 32 

Addition  of  Like  Monomials 36 

Addition  of  Unlike  Monomials 40 

Addition  of  Polynomials 42 

IV.  Subtraction 50 

Subtraction  of  Like  Monomials 50 

Subtraction  of  Unlike  Monomials 52 

Subtraction  of  Polynomials 54 

Parentheses 57 

V.     Multiplication 67 

Multiplication  of  Monomials     ......  69 

Multiplication  of  a  Polynomial  by  a  Monomial        .        .  71 

Multiplication  of  a  Polynomial  by  a  Polynomial      .        .  74 

Type  Forms  in  Multiplication 78 

VI.    Division 92 

Division  of  Monomials     . 94 

Division  of  a  Polynomial  by  a  Monomial          ...  96 

Division  of  a  Polynomial  by  a  Polynomial       ...  99 

VII.     Simple  Equations 104 

The  Solution  of  Problems 116 

Rules  and  Formulas         .        .        .        .        .        .        .  123 

v 


vi  Contents 

PA0B 

VIII.     Factoring .        .        .132 

The  Solution  of  Equations  by  Factoring        .        .        .156 

IX.    Highest  Common  Factor  and  Lowest  Common  Multiple  .  161 

Highest  Common  Factor 161 

Lowest  Common  Multiple 165 

X.     Fractions              .        . 167 

Reduction  of  Fractions 168 

Addition  and  Subtraction  of  Fractions  .         .        .         .179 

Multiplication  of  Fractions 186 

Division  of  Fractions 190 

Complex  Fractions 194 

XL    Equations  Containing  Fractions 197 

Review  of  Fractions  and  Fractional  Equations      .         .  214 

XII.     Ratio  and  Proportion 218 

Properties  of  Proportions 222 

XIII.  Graphs 237 

XIV.  Linear  Simultaneous  Equations 250 

XV.  Square  Root        .                279 

Square  Root  of  Polynomials 280 

Square  Root  of  Arithmetical  Numbers  ....  283 

XVI.  Quadratic  Equations  .        . 290 

Incomplete  Quadratic  Equations   .        .        .        .        .  291 

Complete  Quadratic  Equations 294 

XVII.    Simultaneous  Equations  Involving  Quadratics         .        .  305 

XVIII.    Exponents 315 

XIX.     Radicals ....  331 

Reduction  of  Radicals 333 

Addition  and  Subtraction  of  Radicals  ....  341 

Multiplication  and  Division  of  Radical  Expressions      .  344 

Rationalizing  Factors 849 

Division  of  Polynomial  Radicals 350 

Rationalizing  Denominators 352 

Involution  and  Evolution  of  Radicals    ....  365 


Contents  vii 

PAGE 

XX.    Radical  Equations * .       .        .362 

XXI.     Imaginary  Numbers 367 

XXII.    Quadratic  Equations         • 373 

Complete  Quadratics 374 

Theory  of  Quadratic  Equations 384 

Equations  in  the  Form  of  Quadratics         .        .        .  388 

XXIII.  Simultaneous  Quadratic  Equations 400 

Elimination 412 

XXIV.  Graphical  Solution  of  Equations 418 

XXV.     The  Progressions 427 

Arithmetical  Progression 427 

Geometrical  Progression 434 

Infinite  Geometrical  Series  —  Repeating  Decimals     .  437 

XXVI.     The  Binomial  Formula 442 

XXVII.     Variation 448 

XXVIII.     Logarithms 454 

XXIX.     General  Review 469 

College  Entrance  Examinations 490 

Appendix 494 

Remainder  Theorem 494 

Factor  Theorem 495 

Synthetic  Division 496 

Index 499 


ELEMENTARY   ALGEBRA 

L  INTRODUCTION 

1.  Symbols  Representing  Numbers.  Algebra,  like  arithmetic, 
treats  of  numbers ;  but  in  arithmetic,  numbers  are  usually 
represented  by  means  of  Arabic  numerals,  1,  2,  3,  4,  5,  6,  7,  8, 
9,  0,  while  in  algebra  they  are  represented  also  by  letters,  as 
a,  by  c,  .  .  . ;  .  .      x,  y,  z. 

Numbers  represented  by  letters  are  called  literal  numbers. 

The  student  has  seen  how  the  rules  of  arithmetic  can  be 
abbreviated  by  the  use  of  letters  to  represent  numbers.  When 
a  rule  is  expressed  by  means  of  letters,  the  result  is  a  formula. 

The  rule,  "the  number  of  square  units  in  the  area,  A,  of  a 
rectangle  is  equal  to  the  product  of  the  number  of  units  in  the 
length,  /,  and  the  number  of  units  in  the  width,  w,"  may  be 
expressed  by  means  of  the  formula : 


A  =  I  X  w. 

w 


1.   Find  the  area  of  a  rectangle  whose  Jength 


A=l  xw 
I 


is  6  inches  and  whose  width  is  4  inches.  Rectangle 

Substitute  6  for  I  and  4  for  w  in  the  formula  A  =  I  x  w. 

Thus,  J.  =  Zxto  =  6x4=24,  the  number  of  square,  inches  in  the  area. 

2.  Find  the  area  of  a  rectangle  whose  length  is  12  inches  and 
whose  width  is  9  inches. 

3.  Explain  the  use  of  the  following  formulas  : 

Area  of  a  triangle,  i=|x  ax  b, 

where  b  is  the  number  of  units  in  the  base  and 

a  is  the  number  of  units  in  the  altitude.  Triangle 

Circumference  of  a  circle,  C  =  2  x  irx /?  (tt  =  3.1416). 

Interest  on  money  invested,         /  =  p  X  r  x  t. 


Introduction 


2.  Symbols  Representing  Operations.  The  following  table 
shows  that  the  symbols  of  operation  used  in  algebra  are  the 
same,  with  few  additions,  as  those  used  in  arithmetic : 


Plus 

Minus 

Times 

Divided  by 

3+2 

a  +  b 

3-2 

a  —  b 

3x2 

a  x  b,  a  •  b,  ab 

4-2,   -,4:2 

1   2' 

a  -r-  6,  -  ,  a  :  b 
b 

The  sign  for  equality,  = ,  is  used  as  in  arithmetic. 

Notice  that  while  with  arithmetical  numbers  multiplication 
is  indicated  by  the  sign  x ,  as  3  x  2,  or  3  x  5  X  7,  with  literal 
numbers  the  sign  is  usually  omitted. 

Thus,  3  a,  which  is  read  three  a,  means  3  x  a,  and  3  ab,  which  is  read 
three  ab,  means  3  x  a  x  b. 

3  a  +  5  means  that  3  times  a  is  to  be  increased  by  5,  and  is 
read  three  a  plus  Jive. 

What  does  3  a  —  5  mean  ? 

If  a  =  2,  what  is  the  value  of  3a?  of  3a  +  5?  of3a-5? 

ORAL  EXERCISE 
3.    1.   What  is  meant  by  3a?  by  5  b? 

2.  What  is  meant  by  3 #  +  4  ?  by  3 a;  +  5y? 

3.  If  x  =  2,  what  is  the  value  of  5  x  ? 

4.  If  a  =  2,  and  6  =  4,  what  is  the  value  of  4-^a?  of 
6-s-  a?  of  Sab? 

5.  What  is  meant  by  5  ab  ?  6  xyz  ?    2  mnp  +  rs?    2a-36? 

6.  What  is  meant  by  ab  -s-  c  ?    mnp  h-  r?   xy  -^  s? 

7.  When  x  =  1,  and  y  =  2,  find  the  value  of  3  xy ;  of  x  +  ?/ ; 
of  4  a;  —  2/ ;  of  y  -7-  x. 

8.  Head  3z  +  7  =  12. 

9.  Read  m  -i-  n  =  5  xy.  In  what  other  way  can  the  same 
statement  be  written  ? 


Introduction  3 

10.  Read  3  x  =  x  -f-  x  -f  x. 

11.  Read2a6c  +  6;  2ttE;  I  =  prt. 

12.  Read  3 a;  —  1=5;  2x  —  5y  =  ab. 

13.  Reada  +  6  —  a6+^- 

14.  What  operation  of  arithmetic  is  suggested  by  each  of 
the  following  words  :  sum  ?  quotient  ?  product  ?  difference  ? 

EXERCISE 

4.    Write,  using  proper  algebraic  symbols : 

1.  The  sum  of  2  times  a  increased  by  6. 

2.  The  sum  of  7  a  and  2  b. 

3.  a  times  b  times  c. 

4.  Two  times  x  diminished  by  c. 

5.  The  sum  of  a  and  two  times  b. 

6.  Indicate  that  two  times  some  number,  x,  increased  by  5 
is  equal  to  25. 

7.  Indicate  the  product  of  the  factors  3,  a,  6,  c. 

8.  Indicate  the  sum  of  3  times  b  and  a  times  x. 

9.  Indicate  the  product  of  r  and  s  divided  by  t. 

10.  Indicate  the  quotient  of  d  and  I  increased  by  /  times  g. 

11.  What  does  2  ab  +  3  equal  when  a  =  3  and  6  =  4? 

12.  Find  the  value  of  3  -f  5  a;  when  x  =  2. 

13.  Find  the  value  of  4  +  5  d  when  d  =  12. 

14.  Find  the  value  of  5  a  -\ h  ^  when  a  =  2. 

a      2 
3  3 

15.  Find  the  value  of  -  +  76  when  a  =  -  and  6  =  2. 

a  2 

16.  If  t  stands  for  tens  and  h  for  hundreds,  what  number 
does  6  ft  +  7  £  +  4  represent  ? 

17.  If  y  stands  for  yards,  /  for  feet,  and  i  for  inches,  how 
many  inches  does  14  y  +  11  /  +  5  i  represent  ?  How  many 
inches  does  19  y  —  16  f  +  2  i  represent  ? 


Introduction 

18.  The  side  of  a  square  is  a  inches. 
What  will  represent  the  distance  around 
it? 

19.  If  the  length  of  a  rectangle  is  I  in- 
ches and  its  width  w  inches,  how  can  you 
express  its  area?  State  this  formula  for 
finding  the  area  of  a  rectangle  as  a  rule. 

Express  the  distance  around  the  rectangle.     State  this  result  in 
the  form  of  a  rule. 

20.  The  sides  of  a  triangle  are  x  inches,  2  x  inches,  and  y 
inches.  What  is  the  perimeter,  that  is,  the  distance  around 
the  triangle  ? 

21.  If  a,  6,  and  c  represent  the  units',  tens',  and  hundreds' 
digits  of  a  number,  how  may  the  number  itself  be  represented  ? 

Suggestion.     543  =  100  •  5  +  10  .  4  +  3. 

22.  Find  the  value  of  2  n  when  n  =  1 ;  2  ;  3  ;  4 ;  5.  Does 
2  n  always  represent  an  even  number  when  n  is  any  integer  ? 

23.  When  n  is  an  integer,  does  2  n  —  2  represent  an  even  or 
an  odd  number  ?  2  n  +  2  ? 

24.  Do  2  n  +  1  and  2  n  —  1  represent  odd  or  even  numbers 
when  n  is  any  integer  ? 

25.  Does  2a4-3a  =  5a  when  a  =  2  ?  when  a  =  3  ?  when 
a  =  any  number  ? 

26.  Does  5  a  —  a  =  4  a  when  a  =  5  ?  when  a  =  7  ?  when 
a  =  any  number  ? 

27.  Does  2x36  =  66  when  6  =  4?  when  6  =  10  ?  when 
6  =  any  number  ? 

28.  Does  10  a  -T-  5  =  2  a  when  a  =  1  ?  when  a  =  8  ?  when 
a  =  any  number  ? 

29.  What  number  multiplied  by  5  equals  25?  What  num- 
ber multiplied  by  7  equals  35  ?  If  5  m  =  35,  what  is  the  value 
of  m ?     If  9 h  =  72,  what  is  the  value  of  h? 


Introduction  5 

30.  In  arithmetic,  to  find  the  percentage  when  the  base  and 
the  rate  are  given,  we  multiply  the  base  by  the  rate.  Express 
this  rule  by  means  of  a  formula  when  b,  r,  and  p  represent  the 
base,  the  rate,  and  the  percentage  respectively. 

5.  Factor.  If  two  or  more  numbers  are  multiplied  together, 
a  product  is  formed  and  the  numbers  are  factors  of  the  product. 

Thus,  7  xy  is  the  product  of  7,2,  and  y\  and  7,  x,  and  y  are  the  factors 
of  the  product. 

6.  Exponent.  To  indicate  that  the  number  a  has  been  used 
as  a  factor  twice  in  forming  the  product  ax  a,  we  write  a2 
instead  of  a  x  a ;  for  a  •  a-a  we  write  a3.  These  are  read  a 
square,  and  a  cube,  respectively.  4  a3  means  4  •  a  •  a  •  a  and  is 
read  four  a  cube.  The  numbers  2  and  3  in  a2  and  a3  are  ex- 
ponents. 

7.  Square  Root  and  Cube  Root.  The  sign  V  indicates  the 
square  root  of  a  number,  that  is,  one  of  the  two  equal  factors 
of  a  number. 

Thus,  Vl6  =  4  ;  Vo*  =  a. 

The  sign  -fy~  indicates  the  cube  root  of  a  number,  that  is, 
one  of  its  three  equal  factors. 
Thus,   \/27  =  3;  Vx?  =  x. 

8.  Symbols  of  Deduction.  In  a  series  of  steps  one  of  which 
is  derived  from  another  the  symbols  of  deduction,  .-.  and  v,  are 
used.     These  symbols  are  read  therefore  and  since  respectively. 

ORAL  EXERCISE 

9.  Read  the  following  : 

1.  5a2  +  7ab  +  2.  5.  3x  =  x  +  x  +  x. 

2.  7  •  a  +  a2  —  2.  6.  ax2  +bx+  c. 

3.  3z2+3a;-±2.  7.  Va+VHa2  +  &2. 

4.  3x+7  =  12.  8.  -.-2.4  =  8,  .-.8-5-2  =  4. 


6  Introduction 

9.   Read  v  12  +  3  =  15,  .-.  15  -  3  =  12. 

10.  What  are  the  factors  of  ab?  of  Sx'h?'?  of  5mnp? 
of  9aW? 

11.  Express  the  square  of  a;  the  cube  of  p. 

12.  Express  the  square  root  of  a ;  of  2  m. 

13.  Express  by  using  exponents  2-2-2;  a  x  a  x  a; 
6- 6  -c-c;  p'p*p'q»q. 

14.  If  the  side  of  a  square  is  a,  what  is  its  area  ? 

15.  If  the  edge  of  a  cube  is  a  inches,  express  the  formula 
for  finding  its  volume.  State  the  formula  for  finding  the  area 
of  its  surface.     State  these  two  formulas  as  rules. 

10.  Some  Simple  Operations.  From  the  method  of  writing 
algebraic  numbers  we  are  justified  in  performing  the  following 
simple  operations : 

(a)  2  x  -+-  3  x  =  5  x  f  or  all  values  of  x. 

For  if  two  times  a  number  is  increased  by  three  times  that 
number,  the  result  is  five  times  the  number.  Compare  this 
with  2x4  +  3x4  =  5x4. 

Similarly,  5x  —  Sx  =  2x,  and  7 x  —  x  =  6 x.  (Note  that  x  is 
the  same  as  1  x.) 

(6)  If  2  is  subtracted  from  5  x  +  2,  the  result  is  5  x. 

For  if  the  sum  of  two  numbers  is  diminished  by  one  of  them, 
the  result  is  the  other. 

(c)  2x36  =  2x3x6  =  6x6  or  6  6. 
This  is  similar  to  2x3x4  =  6x4. 
5-4?/ =  20?/.     6-3a  =  ? 

(d)  2  x  -5-  2  =  x,  f or  if  the  product  of  two  numbers  is  divided 
by  one  of  them,  the  quotient  is  the  other.     Also  16m-j-2=8m. 

ORAL  EXERCISE 

11.  Perform  the  indicated  operations: 

1.  2x3o;  4  x5a;  3-26. 

2.  2a-f-2;  lOy-r-5;  8a-*-4;  16a-*-8j  32m-*-4. 


Introduction  7 

3.  Sab  -T-  a ;  12 xy  -+-  4  y  ;  18 />g  -5-  3  j> ;  25  a&c  -r-  a&. 

4.  4 a;  -f-  2  diminished  by  2  ;  7m  +  3  diminished  by  3. 

5.  Add  3  to  5x  +  3.     (5x+  3  +  3  =  5x  +  6.)     Add  6  to 
8p  +  2. 

6.  What  number  subtracted  from  3  x  4-  7"  will  give  3  a;  ? 

7.  What  must  be  done  to  the  number  3  x  +  5  to  get  3  a;? 

8.  What  must  be  done  to  3  a;  to  get  x  ? 

9.  2a;  +  13a;  =  ?     3  x  +  x  +  5  x  =  ? 

10.  5a;  —  4a;  =  ?     8a;- 2  a;- a;  =  ? 

11.  If  5  a;  =  15,  what  does  x  equal  ? 

12.  7a;  +  2a;-3a;=?     9a;- 3a;+ 5a;=  ? 

13.  10a;-5a;  +  8-3  =  ? 

14.  10a;  —  5a;  +  8a-3a=? 

15.  4aj  +  7a;  +  12-8=? 

16.  4a;  +  7a;  +  126-8&=? 

12.  Equation.  A  statement  expressing  the  equality  of  two 
numbers  is  called  an  equation. 

Thus,  2  x  +  3  =  9  is  an  equation. 

The  two  equal  numbers  are  the  members  of  the  equation. 
The  number  written  at  the  left  of  the  sign  of  equality  is  the 
first  member,  while  the  other  number  is  the  second  member  of 
the  equation. 

13.  Unknown  Number.  A  number  in  the  equation  whose 
value  is  to  be  found  is  the  unknown  number. 

Thus,  in  the  equation,  2a5  +  3  =  9,  2#  +  3is  the  first  member  and  9 
is  the  second  member,     x  is  the  unknown  number. 

That  value  of  the  unknown  number  which,  if  substituted 
for  it  in  the  equation,  will  make  the  two  members  equal  satis- 
fies the  equation. 

Thus,  if  3  is  substituted  for  x  in  2  x  +  3  =  0,  we  have  2  •  3  +  3  =  9  or 
6  +  3  =  9.  Therefore  3  satisfies  the  equation.  Does  2  satisfy  5  x  +  1  =  11? 
Does  3  satisfy  5  x  +  1  =  11  ? 


8  Introduction 

14.  Solving  Equations.  Root.  The  process  of  finding  the 
value  of  the  unknown  number  that  satisfies  the  equation  is 
called  solving  the  equation.  The  value  of  the  unknown  number 
that  satisfies  the  equation  is  the  root  of  the  equation. 

Thus,2  is  the  root  of  3  x  +  1=7  because  3-2  +  1  =  7. 

15.  Principles  used  in  Solving  Equations : 

(a)  If  the  same  number  is  added  to  equal  numbers,  the  resulting 
numbers  are  equal. 

(6)  If  the  same  number  is  subtracted  from  equal  numbers,  the  result- 
ing numbers  are  equal. 

(c)  If  equal  numbers  are  multiplied  by  the  same  number,  the  resulting 
numbers  are  equal. 

(d)  If  equal  numbers  are  divided  by  the  same  number,  the  resulting 
numbers  are  equal. 

Note.     Division  by  zero  is  not  included  in  this  last  statement. 

16.  In  the  solution  of  2  x  -f-  3  =  9,  the  steps  are  as  follows  : 

1.  2x  +  3  =  9. 

2.  .-.  2x  =  6.     Subtract    3   from   both   members  of   the   equation. 

See  §  15,  (6). 

3.  .-.  x  =  3.     Divide  both  members  by  2.     See  §  15,  (d). 

17.  Check.  The  solution  of  an  equation  may  be  checked  by 
putting  the  root  obtained  in  the  place  of  the  unknown  number 
in  the  equation.  When*  this  is  done,  if  the  two  members  are 
equal,  the  solution  is  correct. 

Thus,  to  check  the  answer  3  in  the  solution  of  2  x  +  3  =  9,  put  3  for  x, 
then  2x3  +  3  =  9.     Therefore  the  solution  is  correct. 


EXERCISE 

18.    Solve  the  following  equations,  explaining  each  step  by  the 
statement  of  the  principle  involved.     Check  each  solution. 

1.   3a; -+ 5  =  20.       3.   5x  +  2  =  3.       5.    4  n  +  2  =  6. 

ft.   2x  4-  8  =  13.       4.   3a; -I- 7  =  16.     6.   3  a;  +  2  x  +  8  =  23. 


Introduction  9 

7.  2z  +  l  =  4.  10.   3v  +  6  =  ll.      13.   6#  +  7  =  13. 

8.  5w  +  2  =  52.   -      11.   9a  +  l  =  3.        14.   z  +  l=»6. 

9.  ra  +  l=4.  12.   5  +  6  =  12.         15.   5z  +  5=a+9. 

Solution.       1.        5aj  +  5  =  a;  +  9. 

2.  .•.  4 a;  +  6  =  9.     (Subtracting  x  from  each  member.) 

3.  .*.  4aj  =  4.     (Subtracting  5  from  each  member.) 

4.  .-.  x  =  1.     (Dividing  each  member  by  4.) 
Check.     When  z  =  1,  5  a;  +  5  =  10,  and  a;  +  9  =  10.    Therefore  the 

solution  is  correct. 

16.  3x  +  3  =  x  +  5.  19.   5a  +  3a  +  10  =  22. 

17.  122/  +  3=7?/  +  18.  20.   8a  +  2x+  9  =  2a;  +  20. 

18.  12 y  +  3  =  7i/  +17.  21.   7z  +  16  =  2*  +  3  z  +  40. 

EXERCISE 

19.  Writing  Algebraic  Numbers  and  Making  Equations. 

1.  If  n  stands  for  a  number,  what  will  stand  for  three  times 
this  number? 

2.  If  n  stands  for  a  number,  what  will  stand  for  the  num- 
ber increased  by  3  ? 

3.  If  a;  is  an  integer,  what  will  stand  for  the  next  larger 
integer  ? 

4.  If  a  room  is  /  feet  long,  how  many  inches  long  is  it  ? 

5.  How  would  you  express  /  feet  and  i  inches  in  inches  ? 

6.  Express  p  pounds  and  z  ounces  in  ounces. 

7.  Express  the  result  of  multiplying  a  number  a;  by  3  and 
adding  2  to  the  product. 

8.  Indicate  that  two  times  some  unknown  number  x  in- 
creased by  5  is  equal  to  17. 

9.  Find  the  unknown  number  in  example  8 :    (a)  by  arith- 
metic ;  (b)  by  algebra. 

10.  How  can  two  unknown  numbers  be  expressed  if  one  is 
double  the  other  ? 


10  Introduction 

• 

11.  The  sum  of  two  numbers  is  30,  and  one  of  them  is  twice  as 
large  as  the  other.  Find  the  numbers  by  arithmetical  analysis. 
Also  make  and  solve  an  algebraic  equation  to  find  them. 

Suggestion,     x  and  2  x  may  represent  the  numbers. 

12.  The  sum  of  two  numbers  is  45,  and  one  of  them  exceeds 
the  other  by  5.  What  are  the  numbers  ?  Solve  first  by  arith- 
metic, then  by  algebra. 

Notice  how  much  easier  it  is  to  solve  examples  9,  11,  and  12 
by  algebra  than  by  arithmetic. 

13.  Five  times  a  certain  number,  increased  by  2,  is  equal  to 
the  result  obtained  by  multiplying  the  same  number  by  3  and 
adding  14  to  the  product.     Find  the  number. 

Solution.     Let  x  =  the  required  number. 

Hence  5  x  +  2  =  the  result  of  multiplying  the  number  by  5,  and 

adding  2  to  the  result, 

and  Sx  +  14  =  .  .  .  (Let  the  student  complete  the  statement.) 

Then  5a;  +  2  =  3x  +  14.     (By  the  conditions  of  the  problem.) 

.-.  5x  =  Sx+  12.     (Why?) 

.-.  2  a  =12.     (Why?) 

.♦.  x  =  6. 

Therefore  the  required  number  is  6. 

Check.    In  checking  the  solution  of  this  problem,  it  will  not  do  to 

substitute  6  for  x  in  the  equation,  for  an  error  might  have  occurred  in 

forming  the  equation.     The  answer  should  be  substituted  in  the  original 

problem. 

EXERCISE 

20.  Make  and  solve  equations  for  the  following  problems.  Check 
each  result  by  seeing  if  it  satisfies  the  conditions  of  the  problem. 

1.  If  a  certain  number  is  multiplied  by  7  and  the  product 
is  increased  by  5,  the  result  is  equal  to  the  original  number 
increased  by  83.     Find  the  number. 

Solution.     Let  x  =  the  number. 

Hence  7  x  +  5  =  7  times  the  number  increased  by  5, 
and  x  +  83  =  the  number  increased  by  83. 
Then  7  x  +  5  =  x  +  83.     (By  the  conditions  of  the  problem.) 
.-.  7z  =  x  +  78.    (Why?) 


Introduction  11 

.-.6  x  =  78.     (Why?) 
.-.  x  =  13.     (Why?) 
Therefore  13  is  the  required  number. 

Check.  7  x  13  +  5  =  96,  and  13  +  83  =  96.  Therefore  13  is  the  num- 
ber required  by  the  conditions  of  the  problem. 

2.  If  two  times  a  certain  number  is  increased  by  6,  the 
result  is  equal  to  the  sum  of  the  original  number  and  9.  Find 
the  number. 

3.  Find  three  numbers  of  which  the  second  is  double  the 
first,  and  the  third  exceeds  the  first  by  8,  their  sum  being  44. 

4.  The  sum  of  three  numbers  is  24.  The  second  is  double 
the  first,  and  the  third  equals  the  sum  of  the  other  two.  Find 
the  numbers. 

5.  Two  men  have  together  $68.  One  of  them  has  $2 
more  than  twice  as  much  as  the  other.  How  many  dollars 
has  each  ? 

Solution.     1.    Let  x  =  the  number  of  dollars  one  man  has. 
Hence  2  x  +  2  =  the  number  of  dollars  the  other  has, 
and  x  4-  2  x  +  2  =  the  number  of  dollars  both  have. 
Then  x  -4-  2  x  +  2  =  68.     (By  the  conditions  of  the  problem.) 
or3x  +  2  =  68.     (Why?) 
.-.3x=66.     (Why?) 
.-.a  =  22.     (Why?) 
Therefore  one  man  has  §22  and  the  other  man  has  2  x  $22  + $2,  or  $46. 
Check.     Let  the  student  check  the  problem. 

The  student  should  notice  that  x  was  not  used  to  represent  one  man's 
money,  but  the  number  of  dollars  he  had.  The  dollar  sign  is  not  to  be 
placed  with  any  of  the  numbers  in  the  equation.  The  equation  is  ex- 
pressed in  abstract  numbers. 

6.  The  cost  of  a  horse  is  two  times  the  cost  of  a  cow ;  the 
cost  of  a  cow  is  five  times  the  cost  of  a  sheep.  Find  the  cost 
of  each  if  a  horse,  a  cow,  and  a  sheep  together  cost  $  208. 

Suggestion.    Let  x  =  the  number  of  dollars  one  sheep  costs. 

7.  Divide  $55  between  A  and  B  so  that  A  shall  have  $5 
more  than  four  times  as  much  as  B. 

Suggestion.    Let  x  =  the  number  of  dollars  B  has. 


12  Introduction 

8.   The  sum  of  the  angles  of  a  triangle  is  180°.     How  many- 
degrees  are  there  in  each  angle  if  the  largest  angle  is  three 
times  as  large  as  the  smallest  and  the  other  is  twice 
as  large  as  the  smallest  ? 

Suggestion.     Let  x  =  number  of  degrees  in  the  smallest 


9.   The  sum  of  the  lengths  of  the  three  sides  of 
a  triangle  is  17  inches.     The  second  side  is  two 
inches  longer  than  the  shortest,  and  the  third  is 
twice  as  long  as  the  shortest.     Find  the  lengths  of  the  sides. 
Suggestion.     Let  x  =  the  number  of  inches  in  the  shortest  side. 

10.  A  piece  of  rope  106  inches  long  is  to  be  cut  into  two 
parts  so  that  one  part  shall  be  10  inches  more  than  twice  as 
long  as  the  other.     How  long  will  each  part  be  ? 

11.  Henry  is  5  years  older  than  James,  and  the  sum  of 
their  ages  is  37.     Find  the  age  of  each. 

12.  If  f  of  a  number  is  72,  what  is  the  number  ? 
Solution.     Let  x  =  the  number. 

■x  =  72. 

x  =  96.     (Dividing  both  numbers  of  the  equation  by  f .) 

13.  The  sum  of  the  ages  of  three  boys  is  38  years.  The 
youngest  is  |  of  the  age  of  the  oldest  and  3  years  younger  than 
the  second.     How  old  is  each  boy  ? 

Suggestion.     Let  x  =  the  number  of  years  in  the  age  of  the  oldest. 
.-.  x  +  f  x  +  f  x  +  3  =  38.  Explain. 

14.  If  an  automobile  after  being  reduced  25  %  in  price  costs 
$  900,  what  was  its  original  cost  ? 

Suggestion,     x  -  \  x  =  900.     (Why  '.') 

15.  A  salesman  earned  $  20  at  2  %  commission.  Find  the 
amount  of  his  sales. 

Suggestion.     .02  x  =  20. 

16.  A  man  bought  the  same  number  each  of  1^,  2^,  and  4P 
stamps  for  70  £     How  many  of  each  kind  did  he  buy  ? 


Introduction  13 

17.  If  a  debt  of  $  144  is  paid  by  using  the  same  number 
each  of  $  1,  $  2,  $  5,  and  $  10  bills,  how  many  of  each  kind  of 
bills  is  used  ? 

18.  At  an' election  there  were  two  candidates  for  the  office 
of  mayor.  They  together  received  2360  votes.  If  one  candi- 
date was  defeated  by  328  votes,  how  many  votes  did  each 
receive  ? 

19.  At  an  election  there  were  three  candidates  A,  B,  and  C 
for  a  certain  office.  They  together  received  3447  votes.  If  A 
received  twice  as  many  as  B,  and  C  195  more  than  B,  how 
many  votes  did  each  receive  ? 

20.  If  a  field  requires  36  pounds  of  nitrogen  for  fertilization, 
how  much  nitrate  of  soda  containing  18  %  of  nitrogen  will  be 
needed  ? 

21.  In  an  algebra  class  there  are  24  pupils.  If  there  are 
6  more  girls  than  boys  in  the  class,  how  many  boys  are  there? 

22.  A  man  buys  twice  as  much  hard  coal  as  soft  coal  and 
pays  $  108.  If  hard  coal  is  $  7.50  a  ton  and  soft  coal  is  $  3, 
how  many  tons  of  each  does  he  buy  ? 

23.  Two  trains  leave  Buffalo  at  the  same  time  going  in 
opposite  directions.  One  travels  50  miles  an  hour  and  the 
other  40  miles  an  hour.  In  how  many  hours  will  they  be  630 
miles  apart  ? 

24.  Two  trains  leave  Buffalo  at  the  same  time  going  in  the 
same  direction.  One  travels  45  miles  an  hour  and  the  other 
38  miles.     In  how  many  hours  will  they  be  35  miles  apart  ? 

25.  A  merchant's  profits  for  the  second  year  increased  25  % 
over  the  first  year's  profits.  If  the  total  profits  for  the  two 
years  are  $  7623,  how  much  are  the  profits  for  each  year  ? 

Solution.      Let  x  =  number  of  dollars  profit  the  first  year. 

Hence  x  +  \  x  =  number  of  dollars  profit  the  second  year. 
Then  x  +  x  +  |x  =  7623, 

or  fa  =  7623.     (Why?) 
.  •.  x  =  3388,  the  number  of  dollars  profit  the  first  year. 


14  Introduction 

26.  A  workman's  weekly  expenses  are  f  of  his  wages.  How 
much  does  he  earn  each  week  if  he  has  $  5  left  ? 

27.  Two  pupils  together  solve  28  algebra  problems.  One 
of  them  solves  J  as  many  as  the  other.  How  many  problems 
does  each  one  solve  ? 

28.  A  rectangular  field  is  f  as  wide  as  it  is  long  and  its 
perimeter  is  40  rods.     Find  the  length  and  the  width. 

29.  Divide  90  into  two  such  parts  that  one  part  equals  twice 
the  other. 

30.  A  farmer  raised  3000  bushels  of  corn,  wheat,  and  oats. 
If  he  raised  3  times  as  much  corn  as  wheat  and  twice  as  much 
oats  as  wheat,  how  many  bushels  of  each  did  he  raise  ? 

31.  A  farmer  has  4  times  as  many  hogs  as  cattle  and  twice 
as  many  sheep  as  hogs  and  cattle  together.  If  he  has  210 
animals  in  all,  how  many  of  each  kind  has  he  ? 

32.  Three  newsboys  sold  140  papers.  If  the  first  sold  i  as 
many  as  the  second  and  the  third  twice  as  many  as  the  second, 
how  many  did  each  boy  sell  ? 

33.  A  mason  and  his  helper  together  earn  $  6  a  day.  If 
the  helper  earns  i  as  much  as  the  mason,  how  much  does  each 
receive  ? 

34.  A  baseball  team  won  12  games,  which  was  f  of  the  num- 
ber of  games  played.     How  many  games  were  played  ? 

35.  A  boy  bought  a  ball,  a  bat,  and  a  glove  for  $  2.50.  The 
ball  cost  f  as  much  as  the  glove  and  the  bat  -|  as  much  as  the 
ball.     How  much  did  each  cost  ? 


IL    POSITIVE  AND  NEGATIVE  NUMBERS 


21.  The  first  numbers  with  which  we  became  acquainted 
were  the  whole  numbers  used  in  counting,  such  as  1,  2,  3. 
Later  it  was  found  necessary  to  enlarge  our  idea  of  numbers 
and  include  fractions,  as  ^,  ^,  -f,  ^-.  Still  later  it  became 
necessary  to  express  the  value  of  the  square  roots  and  cube 
roots  of  numbers,  as  V2,  V5.  A  still  further  extension  of 
our  number  system  will  now  be  made,  introducing  negative 
numbers. 

22.  A  thermometer  scale  is  marked  as  in  the  figure.  To 
indicate  that  the  temperature  is  10°  below  zero  we  write  —  10°. 
To  indicate  that  the  temperature  is  10°  above  zero  we 
write  +  10°,  or  simply  10°. 

1.  At  noon  on  a  certain  day  the  temperature  was 
8°  above  zero.  At  night  it  had  fallen  6°.  What  was 
the  temperature  at  night  ?  Will  the  equation  8°  —  6° 
=  2°,  indicate  the  method  of  finding  the  answer  ? 

2.  Suppose  the  temperature  is  8°  above  zero  at  noon 
and  falls  12°  in  the  next  six  hours.  What  is  the  tem- 
perature at  6  o'clock  ? 

The  equation,  8°  —  12°  =  —  4°,  indicates  the  method 
of  finding  the  answer. 

3.  If  the  temperature  is  10°  above  zero  in  the  morn- 
ing and  rises  15°  during  the  forenoon,  what  is  the  tem- 
perature at  noon  ?  , 

10°  +  15°  =  25°. 

4.  If  the  temperature  is  10°  below  zero  in  the  morning  and 
rises  15°  in  the  forenoon,  what  is  the  temperature  at  noon  ? 

-  10°  +  15°  =  5°. 
15 


J0Q 

90; 

- 

. 

[80 

to; 

JO 

60J 

ii° 

»] 

L«> 

1£ 

12 

10J 

• 

16  Positive  and  Negative  Numbers 

ORAL  EXERCISE 

23.  Explain  and  give  the  answers  to  the  following  : 

1.  5° +  7°  =  ?  6.  -3°  +  l°  =  ? 

2.  -  3°  +  5°  =  ?  7.  8°  -  5°  =  ? 

3.  -10° +  7°  =  ?  8.  10° -12°  =  ? 

4.  -8° +  8°  =  ?  9.  18° -30°  =  ? 

5.  7°  -9°=?  10.  -5° -2°=? 

24.  An  Extension  of  Subtraction.  In  arithmetic  the  subtra- 
hend must  not  be  larger  than  the  minuend.  Such  an  opera- 
tion as  8  —  12  has  no  arithmetical  meaning,  for  we  cannot 
subtract  from  a  number  more  units  than  the  number  contains. 
In  algebra,  however,  we  do  subtract  a  larger  number  from  a 
smaller  number,  and  such  subtractions '  give  rise  to  negative 
numbers. 

Thus,  8-  12  =  8-8-4  =  0-  4,  which  we  write  -  4. 
Also,     5-6  =  5-5-1  =  0-1  or  -1. 

25.  Positive  and  Negative  Numbers.  There  are  many  pairs 
of  opposite  numbers  similar  to  the  numbers  of  the  thermome- 
ter scale.  The  fact  that  numbers  are  so  related  to  each  other 
can  be  conveniently  represented  by  the  use  of  the  signs  + 
and  — .  When  thus  used  to  represent  the  quality  of  a  number, 
these  signs  are  read  positive  and  negative  respectively.  Thus, 
+  5  is  read  positive  Jive  and  —  7  is  read  negative  seven.  Num- 
bers preceded  by  the  sign  +  to  indicate  the  quality  of  the 
number  are  positive  numbers ;  numbers  preceded  by  the  sign 
—  to  indicate  the  quality  of  the  number  are  negative  num- 
bers. 

The  student  will  note  that  each  of  the  signs  +  and  —  may  have  two 
distinct  uses;  they  may  indicate  the  operations  of  addition  and  subtrac- 
tion, or  they  may  indicate  the  quality  of  a  number. 

26.  We  usually  omit  the  positive  sign  before  positive  num- 
bers,  writing   and   reading  them    exactly  as   in  arithmetic. 


Positive  and  Negative  Numbers  17 

Sometimes,  however,  for  emphasis  or  for  contrast,  we  write 
the  sign  +  before  a  positive  number,  as  (+  5)  or  +  5.  The 
negative  sign  before  a  negative  number  is  never  omitted.  To 
show  that  these  signs  are  quality  signs,  and  not  operation 
signs,  we  often  write  such  numbers  within  a  parenthesis,  thus 
(—  3)  +  (+  5),  read  negative  3  plus  positive  5. 

ORAL  EXERCISE 

27.  Read  the  following,  using  "positive"  and  "negative"  as 
the  names  of  these  signs  when  they  indicate  quality. 

1.  (_3)+2  +  (-3);  -3  +  2 +(-3). 

2.  -3  +  5;  (-3)+ 5;  5+(-3). 
3.-7-4.  8.   23° +(-4°). 

4.  (_2)(-3)  +  4.  9.  15^+(-5^). 

5.  (—a)+ &+(—«).  10.  —$40  +  $17. 

6.  x+(—  y)+y.  11.  —  2x—  3a+(—  2x). 

7.  ra  —  n+(—  m)+a.  12.  5  +(—  7)-  a(—  b). 

13.  If  we  consider  north  positive,  what  should  we  consider 
south  ?  If  rising  temperature  is  positive,  what  kind  of  tem- 
perature is  negative  ? 

14.  What, signs  would  you  associate  with  each  of  the  fol- 
lowing: (1)  Money  earned  and  money  spent?  (2)  A  man's 
property  and  his  debts  ?  (3)  Distance  up  and  distance  down  ? 
(4)  Distance  to  the  right  and  distance  to  the  left  ? 

28.  The  Algebraic  Number  Scale.  Draw  a  straight  line  and 
divide  it  into  spaces  of  equal  length.  Select  some  point  as 
zero  near  the  center  and  name  the  other  points  of  division  as 
indicated.  This  arrangement  of  numbers  on  a  line  is  the 
algebraic  number  scale.     (See  figure,  page  18.) 

Just  as  the  arithmetical  number  scale  (that  part  of  the  alge- 
braic scale  that  extends  from  0  to  the  right)  is  conceived  as 
extending  indefinitely  to  the  right,  so  the  negative  numbers  of 
the  algebraic  scale  extend  from  0  indefinitely  to  the  left. 


18  Positive  and  Negative  Numbers 

29.  Algebraic  Numbers.  The  positive  and  negative  numbers 
together  form  the  system  of  algebraic  numbers,  or  signed  num- 
bers. 

-9-8-7-6-5-4-3-2-1        0        1        2        3        I        5        6        1        8        \ 
I 1 1 1 1 1 1 1 1 1 1 1 1 I iii. 

30.  Addition  of  Signed  Numbers  on  the  Number  Scale. 

1.  To  add  3  and  5  on  the  number  scale,  begin  at  3  and  count 
5  spaces  to  the  right,  arriving  at  the  point  8.  This  gives  the 
result  3  +  5  =  8. 

2.  To  add  (—3)  and  5,  begin  at  —  3  and  count  5  spaces  to 
the  right,  arriving  at  the  point  2.     .-.  — 3  +  5  =  2. 

3.  Arithmetical  numbers  can  be  added  in  any  order ;  thus, 
3  +  2  =  2  +  3.  We  shall  assume  that  the  order  of  adding 
algebraic  numbers  may  be  changed  in  the  same  way ;  thus 
—  3  +  5=5+(—  3).  This  suggests  that  we  may  add  a  negative 
number  by  counting  to  the  left  on  the  number  scale.  To  verify 
this  begin  at  5  and  count  3  spaces  to  the  left,  arriving  at  the 
point  2. 

Similarly,  8  +(-  3)=  5  and  5  +(-  7)  =  -  2.     Why? 

31.  These  considerations  justify  the  following  rules  for 
adding  on  the  number  scale : 

1.  To  add  any  positive  number,  b,  to  any  number,  n,  begin  at  n  and 
count  b  spaces  to  the  right. 

2.  To  add  any  negative  number,  —  c,  to  any  number,  n,  begin  at  n 
and  count  c  spaces  to  the  left. 

EXERCISE 

32.  Verify  the  answers  given  in  examples  1  to  10,  using  the 
above  rule,  with  a  number  scale : 

1.  ;;  +  5  =  8.  3.   7+(-5)=2. 

2.  _4  +  8  =  4.  4.    -7+5=-2. 


Positive  and  Negative  Numbers  19 

5.  5+(-6)=-l.  8.   9+(-3)  =  6. 

6.  -3+(-4)  =  -7.  9.    7+(-8)  =  -L 
7.-5  +  6  =  1.                            10.-5  +  8  =  3. 

Find  the  answers  to  examples  11  to  16  by  the  use  of  the  number 
scale  : 

11.  2+(-5)+(-l).  14.    -7  +  5  +  3+(-l). 

12.  (_8)+7+(-l).  15.    -5  +  5^6^-(-6). 

13.  4+(-3)+2.  16.    _7  +  (-3)+7+(-7). 

33.  The  essential  difference  between  positive  and  negative 
numbers  is  that  they  are  opposite  quantities.  In  adding  a  posi- 
tive number  we  count  to  the  right ;  in  adding  a  negative  num- 
ber we  count  to  the  left.  Any  number  of  negative  units  added 
to  the  same  number  of  positive  units  gives  zero.  If,  in  adding 
a  positive  and  a  negative  number,  the  number  of  positive  units 
exceeds  the  number  of  negative  units,  the  sum  is  a  positive 
number,  but  if  the  number  of  negative  units  exceeds  the  num- 
ber of  positive  units,  the  result  is  a  negative  number. 

EXERCISE 

34.  1.    $  10  gained  and  $  12  lost  results  in  an  actual  loss  of 

$2,  or  $10 +(-$12)=- $2. 

2.  Indicate  by  the  addition  of  signed  numbers  that  a  boy 
has  $4  and  owes  $5. 

3.  Indicate  the  change  in  a  man's  finances,  if  he  spends  $10 
in  the  morning  and  earns  $12  in  the  afternoon. 

4.  Indicate  by  adding  signed  numbers  that  a  boy  won  12 
points  in  a  game  and  was  penalized  3  points.  What  is  his 
score  ? 

5.  In  three  plays  a  football  team  gains  7  yards,  is  penalized 
15  yards,  and  gains  21  "yards.  Show,  by  adding  signed  num- 
bers, the  net  result  of  the  three  plays. 


20  Positive  and  Negative  Numbers 

6.  How  does  the  addition  of  a  negative  number  compare 
with  the  subtraction  of  a  positive  number  containing  the  same 
number  of  units  ?  Illustrate  the  answer,  using  8—5  and 
8  +  (—  5).     Give  another  similar  illustration. 

35.  Absolute  Value.  The  value  of  a  number  without  its  sign 
is  its  absolute  value.  The  absolute  values  of  —  2,  —  3,  3,  5 
are  respectively  2,  3,  3,  and  5. 

ADDITION  OF  SIGNED  NUMBERS 

36.  The  rules  given  in  §  31  for  adding  positive  and  negative 
numbers  by  means  of  a  number  scale  would  be  neither  convenient 
nor  practical  in  adding  large  numbers,  or  in  adding  fractions. 

Following  are  the  first  six  examples  of  §  32  with  their 
answers : 

1.  3+5  =  8.  4.    (-7) +  5=  -2. 

2.  (-4)  +  8  =  4.  5.    5  +  (-  6)  =  -  1. 

3.  7+(-5)=2.  6.    (-3)  +  (-4)  =  -7. 

37.  By  observing  these  examples,  and  others  of  the  same 
type,  we  may  deduce  the  following  rules : 

1.  To  add  two  positive  numbers,  proceed  as  in  arithmetic.  (Ex- 
ample 1 . ) 

2.  To  add  a  positive  and  a  negative  number,  subtract  the  less  absolute 
value  from  the  greater,  and  prefix  the  sign  of  the  number  having  the 
greater  absolute  value.     (Examples  2,  3,  4,  5.) 

3.  To  add  two  negative  numbers,  add  their  absolute  values  and 
prefix  the  negative  sign.     (Example  6.) 

These  rules  must  be  learned. 

EXERCISE 

38.  Work  out  the  first  Jive  examples  by  the  number  scale  and 
also  by  the  rules.     Solve  the  remaining  examples  by  the  rules. 

1.  3 +(-5).  3.    -7  +  5. 

2.  _8+(-2).  4.    -8  +  8. 


Addition  of  Signed  Numbers  21 


5.    10 +  (-8). 

17.    _8  +  10  +  7+(-3). 

6.    17  +  (-20). 

18.   22  + (-54)+ 7. 

7.    _27  +  30. 

19.   23.1 +  (- 20.5) +  (-1). 

8.    -357  +  (-258). 

20.    .4  +  (-3)  +  (-2).> 

9.   536.5 +  (- 233.25). 

21.    -27  +  (-5)  +  6. 

10.   i+(-.5). 

22.    -5  +  (-7)  +  11. 

11.   2.3 +(-3.4) +5.1. 

23.    12  +  (-2)  +  (-5). 

12.    144+(-23)  +  (-7). 

24.    -l  +  (-l)  +  5. 

13.   468 +  (-298) +  (-200). 

25.    -8  +  (-7)  +  9. 

14.    31.2 +  (-2.01) +  (-1.11). 

26.   357 +  (-252). 

15.   4.312 +(-25) +24. 

27.    -  532  +  (-  5)  +  224. 

16.   3+(-5)  +  (-2)+7. 

28.    75  +  2.3  +  (- 5.2). 

29.    -78  +  37  +  (-24). 

Add  the  following : 

30.    -    5                31.        22 

32.         21                 33.    -12 

3                        -52 

-15                         -    7 

-12                       -31 

-17                        -    5 

-    7                           27 

-    3                           18 

34.  Augustus  Caesar  ruled  the  Roman  Empire  45  years,  be- 
ginning his  reign  31  b.c.  Indicate,  by  the  addition  of  signed 
numbers,  the  end  of  his  reign. 

35.  The  Eoman  historian  Livy  was  born  65  b.c.  and  lived  to 
be  82  years  old.  In  what  year  did  he  die  ?  Indicate  by  using 
signed  numbers. 

39.  When  several  numbers  are  to  be  added,  they  may  be 
added  in  the  order  written ;  or  the  positive  numbers  may  be 
added  by  themselves  and  the  negative  numbers  by  themselves ; 
then  the  two  results  may  be  added. 

Thus,  4+(-3)+8+(-5)=l  +  8+(-5)=9+(-5)=4; 
or  4+(-3)+8+(-5)  =  4  +  8+(-3)  +  (-5)=12+(-8)=4. 


22  Positive  and  Negative  Numbers 

EXERCISE 

40.  Find  the  sum  of: 

1.  3+(-5)+7+(-2).  4.   6.4+5.2+(-2.1)+(-.5). 

2.  18+37  +  (-52)  +  (-80).      5.    -.7  +  3.2  +(-4)  +  .25. 

3.  25+(-6)+14+(-2).       6.    f +  $  +(-  J)+(-/|). 

7.  8+(-6)+5+(-ll). 

8.  If  x  + 6  +(- 1)=  14,  find  a;. 

9.   What  number  added  to  10  will  give  8  ?     If  y  + 10  =  8, 
what  is  the  value  y  ? 

10.  What  number  added  to  -  10  will  give  2  ?  If  y  +(-10) 
=  2,  what  is  the  value  of  y  ? 

11.  If  a  +(—  2)  +4  =  6,  what  is  the  value  of  a? 

12.  A  man  has  $650  in  the  bank,  $45  in  his  pocket,  and 
another  man  owes  him  $135.  He  owes  one  man  $240  and 
another  man  $325.  Indicate  by  addition  of  signed  numbers 
his  financial  standing. 

41.  Algebraic  Sum.  The  result  obtained  by  adding  signed 
numbers  is  the  algebraic  sum. 

SUBTRACTION  OF  SIGNED  NUMBERS 

42.  In  arithmetic,  subtraction  is  defined  as  the  operation  of 
taking  one  number,  the  subtrahend,  from  another  larger  or 
equal  number,  the  minuend.  The  result  of  subtraction  is  the 
difference. 

This  definition  would  mean  nothing  in  such  algebraic  sub- 
tractions as,  5  —  (—  8),  —  2  —  5,  6  —  15.  It  is  therefore  neces- 
sary to  have  a  new  definition  of  subtraction  that  will  apply  to 
signed  numbers. 

The  student  will  recall  the  relation, 

difference  +  subtrahend  =  minuend. 

This  relation  was  used  to  verify  answers  in  subtraction  and  is 
the  basis  of  the  following  definition  of  subtraction : 


Subtraction  of  Signed  Numbers  23 

Subtraction  is  the  process  of  finding  one  of  two  numbers 
when  their  sum,  the  minuend,  and  the  other  number,  the  subtra- 
hend, are  given. 

We  shall  apply  this  definition  to  find  answers  to  a  few 
simple  examples  in  subtraction  and  from  these  results  shall 
construct  rules  for  algebraic  subtraction. 

1.5  —  3=?  By  definition  this  means :  What  number 
added  to  3  will  give  5  ?  We  know  that  3  +  2  =  5  and  there- 
fore 5-3  =  2. 

2.  4  —  6  =  ?  According  to  the  definition,  this  asks  the 
question :  What  number  added  to  6  will  give  4  ?  We  know- 
that  6  +  ( -  2)  =  4,  and  therefore  4  -  6  =  -  2. 

3.  4  — (—  6)=  ?  The  minuend,  4,  is  the  sum  of  two  num- 
bers, and  one  of  the  numbers  is  (—6).  Since  (—6)+ 10  =  4, 
therefore  4  —  (—  6)=  10. 

4.  (—4)— 6  =—10.  Let  the  student  explain  by  using  the 
definition. 

5.  (-4)-(-6)=2.     Why? 

The  student  must  make  sure  that  he  understands  the 
answers  in  the  preceding  examples ;  that  is,  he  must  see  that 
they  satisfy  the  requirements  of  the  definition  of  subtraction. 

43.  The  method  of  subtracting  by  using  the  definition  as  a 
rule  would  not  be  practical.  We  proceed  to  discover  rules 
that  will  simplify  the  process.  Collecting,  for  the  sake  of 
comparison,  the  results  of  §  42,  we  have, 

1.  5 -(+3)  =  2.     Compare  this  with  5+ (-3)  =  2. 

2.  4  - (+  6)  =  -  2.     Compare  this  with  4  +  (-  6)=  -  2. 

3.  4 -(-6)=  10.     Compare  this  with  4+  (+6)=  10. 

4.  -4 -(+6)  = -10.  Compare  this  with  _4+(-6)  = 
-10. 

5.  _4_(_6)=2.     Comparethis  with  _4+(+6)  =  2. 


24  Positive  and  Negative  Numbers 

44.  These  comparisons  indicate  that  we  can  change  any 
subtraction  to  an  addition  by  changing  the  sign  of  the  subtra- 
hend.    Therefore  we  have  the  following  rule  : 

To  subtract  one  signed  number  from  another,  change  the  sign  of  the 
subtrahend  and  add  the  resulting  number  to  the  minuend. 

Examples 

1.  13  -(-4)=  13  +  4  =  17. 

2.  3-(-4)=3+4  =  7. 

3.  4-(-10)=4  +  10  =  14. 

4.  _5-(+3)  =  -5+(-3)  =  -8. 

5.  8-(-3)-(-2)=8  +  3+2  =  13. 

6.  5_(_3)  +  (_2)=5  +  3+(-2)=6. 

7.  243  -  (-  500)  =  243  +  500  =  743. 

8.  -350 -(-250)= -350 +  250  =  -100. 

EXERCISE 

45.  Find  the  value  of : 

1.  7 -(-7).  5.  123 -(-21).  9.  -22- (-3). 

2.  3-10.  6.  2.75 -(-|).  10.  3.5 -(-2.2). 

3.  _5-3.  7.  -37-15.  11.  0-(-2). 

4.  17 -(-3).  8.  .02 -(-.1).  12.  0-(-3). 

13.  2 -(-.2).  22.  -15+(?)  =  12. 

14.  _4-4-4.  23.  15-(?)=20. 

15.  (-4)-(-4)-(-4).  24.  (?)- 10  =  17. 
16-  |-f-(-|).  25.  (?)-(- 10)=  17. 

17.  .5+(-i)-.5.  26.  (?)-(-13)=8. 

18.  17 -(-3)- 3.  27.  (?)-(- 5)=  0. 

19.  0-(-3)+2  +  16.  ,  28.  0-(-10). 

20.  -17  +(-3)-  16.  29.  _(-4)-(-4)-(-4). 

21.  15+(?)=12.  30.  -(-5). 


Multiplication  of  Signed  Numbers  25 

31.  Subtract  -  7  from  15.     Subtract  218.94  from  -123.011. 

32.  Subtract  12  from  -  26.     Subtract  -  5132  from  -  2341. 

33.  What  number  increased  by  15.123  equals  3.102  ? 

34.  The  minuend  is  8.231,  the  subtrahend  is  12.0003 ;  find 
the  difference. 

35.  The  subtrahend  is  —54.265  and  the  difference  is 
-  2.1981 ;  find  the  minuend. 

MULTIPLICATION  OF  SIGNED  NUMBERS 

46.  The  result  of  multiplication  is  the  product.  The  num- 
bers multiplied  are  the  factors  of  the  product. 

47.  There  are  four  cases  of  multiplication  of  signed  numbers. 
The  indicated  multiplication  3x4   is   to  be  read  "  three 

times  four  "  ;  that  is,  the  first  factor  is  taken  as  the  multiplier. 

1.  In  arithmetic,  3x4  means  that  4  is  to  be  added  3  times. 
Thus,       3  x4  =4+4  +  4  =  12,  or  (+  a)  •  (+&)=+«&. 

2.  Similarly,  3x(— 4)  means  that  (—4)  is  to  be  added 
3  times. 

Thus,3x(-4)  =  (-4)  +  (-4)  +  (-4)  =  -12,or(  +  a)  ■  (-b)  =  -ab. 

3.  Since  to  multiply  by  +-3  we  add  the  multiplicand  3 
times,  it  is  reasonable  to  assume  that  to  multiply  by  —  3  we 
subtract  the  multiplicand  3  times ;  that  is,  (—  3)  x  4  means 
that  4  is  to  be  subtracted  3  times. 

Thus,  (_3)x4=-4-4-4=-12,  or(-a)  •  (+&)  =  _«&. 

4.  As  in  3,  (-3)x  (-4)  =  -(-  4)-(-  4)-(-  4) 

=  4  +  4  +  4  =  12,  or  (-  a)  .  (-  b)=ab. 

48.  Collecting  the  results  in  these  four  cases,  we  have  all 
possible  combinations  of  signs  for  two  factors. 

(+3)x(+4)=12,  or  (+a)x(+&)  =  +  a&. 
(+3)x(-4)  =  -12,  or  (a)x(-  b)  =  -ab. 
(-  3)  x  (+  4)  =  -  12,  or  (-  a)  x  (+  b)=  -  ab. 
(-  3)  x  (-  4)=  12,  or  (-  a)  x  (-  6)  =  +a&. 


26  Positive  and  Negative  Numbers 

49.  The  preceding  equations  give,  in  algebraic  symbols,  the 
law  of  signs  for  multiplication,  and  the  method  of  multiplying 
two  signed  numbers. 

To  find  the  product  of  two  signed  numbers  : 

1.  Find  the  product  of  the  absolute  values  of  the  two  numbers. 

2.  Make  the  sign  of  the  product  positive  if  the  two  factors  have  like 
signs,  and  negative  if  they  have  unlike  signs. 

Examples 

1.  3x(— 7)=  — 21.  What  is  the  absolute  value  of  the 
product  ?     Why  is  the  sign  of  the  product  negative  ? 

2.  (—  8)  x  (—  7)  =  56.  Why  is  the  sign  of  the  product  posi- 
tive ? 

3.  (-2)x(-5)x(-2)  =  10x(-2)=-20.     Explain. 

ORAL  EXERCISE 

50.  Find  the  value  of: 

1.  -3x6;   -3  X  6a. 

2.  -3  x(-6);  _3ax(-6). 

3.  7  x(-'3);  7  x(-Sn). 

4.  -10x2.5;  -  10  a  x  2.5. 

5.  12x(-7);  12x(-7s). 

6.  (-3)x(-22);  (-3)x(-22m). 

7.  8  x  (  -  6)  X  5. 

8.  12x(-2)x(-.3). 

9.  (_6)x5x(-J)x(-4). 

10.    (-2)x(-2)x(-2)x(-2)x(-2). 

11.  Given  the  numbers  2,  5,  —3,  —2,  i,  -5,  —.25;  tell  at 
sight  the  product  of  each  number  multiplied  by  each  one  that 
comes  after  it. 

12.  What  sign  has  the  product  when  three  negative  num- 
bers are  multiplied  together  ?  four  negative  numbers  ?  five  ? 
Give  an  answer  that  will  apply  to  all  cases. 


Division  of  Signed  Numbers  27 

DIVISION  OF  SIGNED  NUMBERS 

51.  Division  is  the  process  of  finding  one  of  two  factors 
when  their  product  and  the  other  factor  are  given. 

The  result  of  division  is  the  quotient. 

52.  From  the  definition  of  division  we  derive  the  following : 

1.  Since  (+  7)  •  (+  3)=  +  21,  therefore  (+21)-h(+7)  =  +3. 

2.  Since  (+  7)  •  (-  3)  =  -  21,  therefore  (-  21)-(+  7)  =  -3. 

A  negative  number  divided  by  a  positive  number  gives  a  negative 
quotient. 

3.  Since  (-7)  .  (-3)  =  (+21),  therefore  (+21)-s-(-7)=  -3. 
A  positive  number  divided  by  a  negative  number  gives  a  negative 

quotient. 

4.  Since  (-  7)  .  (+3)  =  -  21,  therefore  (-21)  +  (-7)  =+3. 
A  negative  number  divided  by  a  negative  number  gives  a  positive 

quotient. 

53.  Generalizing  these  results,  we  have  the  rule  for  dividing 
signed  numbers. 

To  divide  one  signed  number  by  another  : 

1.  Find  the  quotient  of  the  absolute  value  of  the  dividend  divided  by 
the  absolute  value  of  the  divisor. 

2.  Make  the  sign  of  the  quotient  positive  if  the  dividend  and  divisor 
have  like  signs  and  negative  if  they  have  unlike  signs. 

Examples 

1.  -12-*- 3  =  -4. 

2.  -12a-=-3  =  -4a. 

4.  -10  -K-5)=  2. 

5.  (-10a)-(-5«)=2. 

6.  (_8)x(-2)-(-4)=16--(-4)  =  -4. 

7.  (-2)2-*-23  =  4-f-8  =  ^. 

8.  (-2)3--22  =  -8h-4  =  -2. 


28  Positive  and  Negative  Numbers 

ORAL  EXERCISE 

54.  Perform  the  operations  indicated  : 

1.  _l2-=-(_3).  5.-1-5-1  9.  _7a+(-7). 

2.  (-12) +4.  6.   3  a  -=-3.  10.  -32 +(-2)4, 

3.  16  +(_4).  7.    -7o  +  a.  11.  _!+(_$). 

4.  -28  +  7.  8.    -7a-=-7.  12.  12  a? +(-4). 

13.  aft -s- (-a).  22.  -  39  h-  13  X  (-  3). 

14.  7x(-6)-(-3).  *  23.  43-=-(-4)2. 

15.  21-(-7)-(-3).  24.  (-3)2-=-33. 

16.  7x(-2r)+2.  25.  (_3)3-=-(-3)2. 

17.  12-(-4)x(-l)3.  26.  54+(-9)x(-6). 

18.  -2x(-3)  +  (-l).  27.  (-2)3X(-3)X(-1). 

19.  (-j)+2.  28.  (_5a6)  +  (-a&). 

20.  -5a-j-(-l)+(-5a).  29.  (-  2)2  +(-  3)2-(-  4). 

21.  _32-(-8)  +  (-4).  30.  -2-f 

55.  Order  of  Operations.  In  a  chain  of  operations  involving 
the  signs,  +-,  — ,  x,  +,  the  numbers  connected  by  the  signs 
X  and  -7-  must  be  operated  upon  first  from  left  to  right  in  the 
order  in  which  they  occur.  The  results  thus  obtained  should 
be  added  and  subtracted  as  indicated  by  the  signs  +  and  — . 

Thus,  3  +  4x2-6-r3=3  +  8-2  =  9. 
Also  3  +  6-^2x5  +  7  =  3  +  3x5  +  7. 
=  3+15  +  7. 
.         =25. 

EXERCISE 

56.  Find  the  value  of: 

1.  2x(-3)+-(-5)x2.  3.    _3x(-4)  +  (-5)-70. 

2.  -7-(-6)+-(-12).  4.   0-2x(-3)+-7-(-l). 

5.  -lx(-2)x(-3)+6x(-2). 

6.  -5  +  3x7-(-5)x(-4). 


Division  of  Signed  Numbers  29 

7.  24  +  8  x2-(-14).  9.    24-3x4-6x5. 

8.  60  -  5  x  3  +  6  -  3.  10.   24-3  +  4-6  +  5. 

11.  24-6  +  3  x  5x4. 

12.  5  +  6  x  7  -  28  x  2  +  3  x  (-  6). 

13.  4  -  3  x  2  +  8  x  (-  2)  +  4  -  (-  1). 

14.  -8x(-2)-15-(-3)  +  7x0. 

15.  0-4x8  +  7x(-2)-(-20). 

16.  15  +  (-3)  +  (-7)2-(-8)2  +  12. 

17.  l20  +  (-3)3+  (-2)  x  8-21-12. 

18.  _15-(-5)  +  8x  (-2) -7  + (-3). 

19.  -  12  x  (-  2)  +  (-  3)  x  8  -  10  x  (-  1). 

20.  0-lOx  (-2)-(-4)  x  8- 7 -(-7). 

21.  12  -  15  -  (-  13)  -  15  -  (-  15). 

22.  12  x  (-  l)-(-  10)  x  (-  1)  -  8  x  (-  l)-(-  6)(-  1), 

If  a  =  6,  6  =  —  5,  c  =  —  3,  d  =  —  i,  e  =  —  -|,  find  the  values 

of  the  expressions  in  examples  23  to  43. 

23.  ab.  30.   ab  —  c3.  37.    be  — be  — 2  a. 

24.  ac\  31.    b  —  ad.  38.    abede. 

25.  36c.  32.    -36  + 2  c.  39.    a2  -  62. 

26.  —  a6c.  33.    ~(ac)—  4^.  40.    62  —  (—  a)2. 

27.  bed.  34.    dc  +  a  +  26.  41.    6a  -56  -3c. 

28.  a  +  be.  35.    6de  +  a  +  6  —  1.     42.   3c  —  2d  +  5  e. 

29.  6  —  c2.  36.    6e  -  6c  +  2  a.         43.    5  a  +  6  —  4  d. 

44.  Does  3a;—  5  =  7  a  —  9  when  x  =  3 ?  when  cc  =  1  ? 

45.  Does  ^-5.t  +  6  =  0  when  a  =  3  ?  when  a  =  2  ? 

46.  Does  X  =  —  1  satisfy  the  equation  x2  —  2x—  3  =  0? 


30  Positive  and  Negative  Numbers 

REVIEW  EXERCISE 

57.  1.  What  quality  signs  would  you  associate  with  each  of 
the  following :  north  latitude,  south  latitude  ?  rising  tem- 
perature, falling  temperature  ?  debts,  credits,  money  lost, 
money  spent,  money  earned,  money  found  ?  a.d.,  b.c.  ?  points 
won  in  a  game,  points  lost,  penalties  ? 

2.  Compare  the  addition  of  a  negative  number  with  the 
subtraction  of  a  positive  number  having  the  same  absolute 
value.     Illustrate. 

3.  Indicate  the  net  result  of  $  10  earned,  $3  spent,  $2 
found,  and  $  2  spent. 

4.  The  temperature  at  8  o'clock  was  28° ;  at  10  o'clock  it 
had  risen  4° ;  at  noon  it  was  5°  warmer  than  at  10  o'clock ;  at 
2  o'clock  it  had  risen  2°  more ;  at  4  o'clock  it  was  3°  colder  than 
at  2  o'clock ;  at  6  it  was  4°  below  the  temperature  at  4  o'clock  ; 
and  at  8  p.m.  it  was  7°  colder  than  at  6  o'clock.  (1)  Indicate 
by  arithmetical  additions  and  subtractions  the  temperature 
at  8  p.m.  (2)  Find  the  same  result  by  addition  of  signed 
numbers. 

5.  If  you  walk  3  miles  south  and  7  miles  north,  how  far 
and  in  what  direction  from  the  starting  point  are  you? 
Indicate  by  adding  signed  numbers. 

6.  How  far  upstream  are  you  if  you  have  rowed  7  miles 
up  and  drifted  2  miles  down  ?  Indicate  the  process  of  finding 
the  answer  in  two  ways. 

7.  Pikes  Peak  is  14,108  feet  above  sea  level.  A  place  in 
Holland  is  161  feet  below  sea  level.  How  much  higher  is  Pikes 
Peak  than  the  place  in  Holland  ?  Indicate  two  ways  of  finding 
the  answer. 

8.  If  a  gasoline  launch  can  run  14  miles  an  hour  in  still 
water,  how  fast  can  it  run  up  a  river  whose  current  flows  4 
miles  an  hour  ?     How  fast  can  it  run  downstream  ? 


Review  Exercise  31 

9.  If  a  person  can  swim  2\  miles  an  hour  in  still  water, 
represent  his  rate  when  swimming  against  a  current  of  3  miles 
an  hour.     Represent  his  rate  downstream. 

10.  The  Roman  Empire  fell  476  a.d.,  622  years  after  the 
fall  of  Carthage.     What  was  the  date  of  the  fall  of  Carthage  ? 

11.  Give  the  rules  for  addition,  subtraction,  multiplication, 
and  division  of  signed  numbers. 

12.  Define  subtraction  ;  define  division. 

13.  What  is  the  basis  of  the  rule  for  subtraction  of  signed 
numbers  ?  of  the  rule  for  division  ? 

14.  What  is  the  absolute  value  of  a  number  ? 

15.  What  is  the  sign  of  (-  l)10  ?  of  (-  l)11  ?     Can  you  give 
an  answer  that  will  apply  to  all  such  examples  ? 

16.  What  is  the  "  order  of  operations  "  ? 

When  a  =  8,  b  =  —  3  and  c  =  —  9,  find  the  value  of: 

17.  a 4-6+  c  21.   a  +  b+c  +  c.       25.   a2  +  b2  +  c. 

18.  a—b  —  c.  22.    b  —  b2.  26.    ab  +  be  +  ac. 

19.  a  —  be.  23.    ab  —  be.  27.    a2  —  ac. 

20.  a  —  b-\-c.  24.    abc  —  c.  28.    —  a  —  b  —  c. 


Ill    ADDITION 

58.  Algebraic  Expression.  A  number  represented  by  alge- 
braic symbols  is  an  algebraic  expression. 

Thus,  2  ab,  5  —  3  ab,  4  +  2  b  are  algebraic  expressions. 

59.  Monomial,  Term.  An  algebraic  expression  the  parts  of 
which  are  not  separated  by  either  of  the  signs  +  or  — ,  is  a 
monomial  or  a  term. 

Thus,  2  ab,  —  xy,3x  +  7  are  monomials. 

60.  Polynomial.  An  algebraic  expression  consisting  of 
two  or  more  terms  is  a  polynomial. 

Thus,  3  ax  —  4  c  +  7  and  m  —  n  +  11  xy  —  16  are  polynomials. 

The  monomials  that  make  up  the  polynomial  are  the  terms 
of  the  polynomial. 

Thus,  3  ax,  —  4  c,  and  7  are  the  terms  of  the  polynomial  3  ax  —  4  c  +  7. 

A  polynomial  of  two  terms  is  a  binomial,  and  one  of  three 
terms  is  a  trinomial. 

Thus,  2  a  +  b  is  a  binomial,  and  ax  —  by  +  c  is  a  trinomial. 

ORAL  EXERCISE 

61.  In  the  following  expressions,  name  (a)  the  monomials, 
(b)  the  binomials,  (c)  the  trinomials,  (d)  the  polynomials: 

1.  3a2a\  4.    -x2-2ax.  7.   \at\ 

2.  4a2  +  z.  5.    4a-r6.  8.    2x2-hSx-l. 

Z.2b  —  c  +  3d.         6.   4  +  a  —  6.  9.   a  -  6  —  c. 

32 


Addition  33 

Name  the  terms  in  each  of  the  following  polynomials : 

1.0.    3a—  b.  12.   |a£2+2a  +  7.    14.   mxn+m-i-n  —  1. 

11.    2a2-3a6+c.        13.    ax2  +  bx  +  c.    15.    -  3  a  -  2  &  -  c. 

62.  Coefficient.  Any  factor  of  a  term,  or  the  product  of  two 
or  more  of  the  factors  of  a  term,  is  the  coefficient  (co-factor)  of 
the  product  of  the  other  factors. 

Thus,  in  2  abc2,  2  is  the  coefficient  of  abc2  ;  2  a  is  the  coefficient  of  &c2, 
etc. 

What  is  the  coefficient  of  xy2  in  5  a;?/2  ?  of  x  ? 

63.  Numerical  Coefficient.  The  numerical  factor  of  a  term  is 
its  numerical  coefficient. 

Thus,  the  numerical  coefficient  of  7  am  is  7.  la  and  7  m  are  literal 
coefficients  of  m  and  a  respectively. 

When  we  speak  of  the  coefficient  of  a  term  we  generally 
mean  the  numerical  coefficient,  including  the  sign  preceding 
the  term. 

Thus,  2  is  the  numerical  coefficient  of  2  abc  and  —  3  is  the  numerical 

coefficient  of  —  3  ax.     Also  -  is  the  numerical  coefficient  of  -  • 
2  2 

What  are  the  coefficients  of  x  and  y  in  the  equation  3  x  4-  4  y 
=  7  ?  What  are  the  coefficients  of  x2  and  x  in  ax2  -\-bx  +  c  = 
0? 

If  no  numerical  coefficient  is  expressed,  the  coefficient  1  is 
understood. 

Thus,  x  is  the  same  as  1  x. 

What  is  the  numerical  coefficient  of  ab2?  of  —  a2? 

64.  Power,  Exponent,  and  Base.  The  product  arising  from 
using  a  number  one  or  more  times  as  a  factor  is  a  power  of  the 
number. 

The  number  written  to  the  right  and  above  another  number 
to  indicate  how  many  times  the  number  is  used  as  a  factor 


34  Addition 

is  the  exponent  (§  6)  of  the  power.     The  repeated  factor  is  the 
base. 

Thus,  a4  means  the  fourth  power  of  a,  often  read  a  fourth  power,  or 
simply  a  fourth.    4  is  the  exponent  of  the  power  and  a  is  the  base. 

a1  means  the  same  as  a.  The  exponent  1  is  never  written,  a2  and  a3 
are  read  "  a  square  "  and  "  a  cube,"  since  if  a  represents  the  length  of 
the  side  of  a  square  or  the  edge  of  a  cube,  a2  and  a8  denote  the  area  of 
the  square  and  the  volume  of  the  cube  respectively. 

65.  The  student,  must  note  carefully  the  difference  between 
coefficient  and  exponent. 

Thus,  3  a  means  3  x  a,  while  a3  means  a-  a-  a,  that  is,  it  means  that 
a  is  used  as  a  factor  three  times. 

ORAL  EXERCISE 

66.  1.  How  would  you  write  3  •  a  •  a  •  b  •  6  using  expo- 
nents ?     5  - a-a-a-x-x-x?     1000? 

2.  How  would  you  write  as  one  term  a+  a  +  a  -f-a? 

Name  the  numerical  coefficients,  the  exponents,  and  the  base  for 
each  exponent  : 

3.  5a2.  6.   4 ran.  9.   3a  —  \ a3. 

4.  2xf.  7.    -3  a5.  10.   a3  -  12  b\ 

5.  x2yn.  8.   x.  11.    —a3. 

Evaluate  (that  is,  find  the  value  of)  the  following  when  a  =  2 
and,  b  =  —  1 : 

12.  ab;  ab2;  abz. 

13.  -2a2b;  -a262;  -  a2b. 

14.  a  +  b  ;  a  -  6  ;  a2  +  62 ;  a3  +  63. 

15.  What  is  the  meaning  of  m4  ?  of  4  m  ? 

16.  Find  the  value  of  a3  when  a  =  2  ;  of  3  a. 

17.  Find  the  value  of  a2  when  a  =  —  2  ;  of  —  a2 ;  of  —  ab. 


Addition  of  Like  Monomials  35 

67.  Similar  and  Dissimilar  Terms.  Terms  that  do  not  differ 
at  all  or  that  differ  only  in  their  numerical  coefficients  are 
like  terms  or  similar  terms. 

Thus,  2  ab,  ab,  and  6  ab  are  similar  terms. 

Terms  that  differ  in  other  respects  than  in  their  numerical 
coefficients  are  unlike  terms  or  dissimilar  terms. 

Thus,  2  ab  and  12  ab2  are  dissimilar  terms.    Why  ? 

ORAL  EXERCISE 

68.  In  the  following  list,  select  all  terms  that  are  similar  to  the 
first;  to  the  second;  to  the  fourth;  to  the  fifth  : 

1.  2a2x.  4.   5x.  7.   5a*x.  10.    — 16  a2x. 

2.  4a6c.  5.  4  ax.  8.    —  3  x.         11.    5  ax2. 

3.  — 4a6c.        6.   ±abc.  9.    ax.  12.   fax. 

ADDITION  OF  LIKE  MONOMIALS 
ORAL  EXERCISE 

69.  Add  the  following  : 

1.   4  and  7  ;  —  2  and  4. 

2.-4  and  7  ;  —  3  and  8  ;  5  and  —  4. 

3.   4  and  -  7  ;  10  and  -  12  ;  -  12  and  10. 

4.-4  and  —  7 ;  —  5  and  —  7  ;   —  3  and  3. 

5.  13  inches  and  5  inches  ;  13  i  and  5 1. 

6.  4  miles  and  7  miles  ;  4  m  and  7  m. 

7.  5  rods,  6  rods,  and  11  rods  ;  5r,  6  r,  and  11  r. 

8.  $  5,  $  7,  $  15  ;  5  d,  7  d,  15  d. 

9.  -11,17,13;  -11a,  17  a,  13  a. 

10.  8,  -9,  -  5;  8»,  -9a?,  -5z. 

11.  m,  10  m,  —  7  m,  —  4  m. 


36  Addition 

Add  the  following  : 

12.  7  ab,  5ab,  —  6  ab,  4  ab. 

13.  3  x  5,  7  x  5,  -  8  x  5,  2  x  5. 

14.  -  9  x  3,  -  4  x  3,  13  x  3,  -  6  x  3. 

70.  These  examples  suggest  the  following  rule : 

To  add  like  monomials,  add  the  numerical  coefficients  and  make  their 
sum  the  coefficient  of  the  common  literal  part. 

In  applying  this  rule,  the  numerical  coefficients  should  be 
added  according  to  the  rules  for  adding  positive  and  negative 
numbers.  The  literal  part  of  each  term  is  thought  of  as  the 
unit  of  addition. 

Terms  may  be  added  in  any  order. 

Examples 

1.  Add— 5  a  and  7  a. 

—  5  +  7=2.     (Adding  numerical  coefficients.) 
.•.  —  5  a  +  7  a  =  2  a. 

2.  4a+(-7a)=-3a. 

3.  ax +(—3ax)-\-(— 5ax)=  —  7  ax. 
Hint.     1  +(-  3)  +  (-  5)  =  -  7. 

ORAL  EXERCISE 

71 .  Add  the  following : 

1.  4  a,  3  a.  9.  3  m,  4  m,  —5  m. 

2.  4  a,  —3  a.  10.  —2x,5x,—7x. 

3.  — 4a,3a.  11.  4r,  —  5r,  6r. 

4.  -4a,  -3a.  12.  -5t,2t,3t. 

5.  lip,  —  7 p.  13.  5ab,  4a6. 

6.  —5  s,  —6  s.  14.  —6xy,  —2xy. 

7.  6n,5n,2n.  15.  7mn,  —  2mn. 

8.  6n,  -5n,  2n.  16.  10x5,8x5. 


Addition  of  Like  Monomials  37 


18.    -  6  x  13,  5  x  13. 

21.  ax*  22.        7  x 

—  asc3  —  3  x 

—  4  aic3  2x 

—  9  ax3  —  5x 


17.   11x7, 

-8 

X7. 

19.        262 

20. 

2c 

-262 

-3c 

62 

-5c 

-362 

-4c 

462 

10  c 

-4aa?  12 


x 


23.  3a2+(-5a2)+(-7a2)+2a2  +  a2  =  ? 

24.  —  5  ax  +(—  3ax)-\-ax  +  5ax  =  ? 

25.  ±d+(-±d)+d+(-2d). 

26.  Solve3a;  +  5a;-8  =  16. 

Solution.    3x  +  5x  —  8  =  16. 

8x-8=16.  (Why?) 

8x  =  24.  (Adding  8  to  both  members  (§  13,  a).) 

x  =  3.  (§13,d.) 

/Sofae  Me  following  equations  : 

27.  5z-3  =  7.  34.  14a;+(-5a;)  =  63  +  (-9). 

28.  4a;+2a;  =  12  +  (-3).  35.  4a;  +  (-3a;)=  10. 

29.  _5y  +  8y  =  7  +  (-3).  36.  8a;  +  (-  4a)  =  -4  +  7. 

30.  8n+(—  3w)+w  =  12.%  37.  3a; -5  =  7. 

31.  15r+(-r)+2r  =  20.  38.  7a;-2  =  5l 

32.  p+  (-p)+3p  =  l.  39.  |a;  +  4=10. 

33.  -3x+10a;  =  12.  40.  4a; +  3  =  13. 

EXERCISE 

72.   Add  the  following  : 

1. 


.2   6 

2. 

-3      a; 

3.       wma; 

4. 

24  r 

-3.1    6 

4      x 

—  2mna; 

-12r 

4      6 

-2.5   2 

—  5  mwa; 

-48r 

-         6 

.08  a; 

—  4  mna; 

122  r 

.086 

12.2  x 

—  7  mnx 

-57r 

Suggestion.     When  adding  several  similar  terms,  we  usually  first  add 
the  positive  numbers,  then  the  negative  numbers,  then  the  results. 


38  Addition 

Add  the  following : 

5.  425  m,  —  321m,  —  m,  —  50  m. 

6.  4 a,  —  9 a,  a,  5 a,  —  6 a,  2a. 

7.  -  250  jpg,  75j>g,  50j9g,  125pg. 

8.  —  10.1a,  .2  a,  —  a,  —2  a,  —  5.1a. 

9.  210/),  -352p,71p,  -83p. 

10.  14.3  g,  -  2.03  q,  17.5  g,  -  .1  q,  q. 

11.  -  5  ab2,  - 11  a&2,  14  ab2,  -  a&2. 

12.  13  r,  -15r,  -r,  73  r,  r. 

13.  a,  —  2  a,  3  a,  —  4  a,  5  a,  —  6  a. 

14.  2  a,  —  4  a,  6  a,  —  8  a,  10  a,  12  a. 

15.  6  a,  .5  a,  —  .Ola;,  -  1.72a. 

16.  -  27  ab,  -  35  ab,  43  a&,  -  20  ab. 

17.  44  xy,  —12xy,  —  24  a?/. 

18.  75,  -  32,  -  70,  23,  -  16.  ' 

19.  784a,  -369  a,  -Ilia,  -53a. 

20.  23  xyz,  —  24  xyz,  —  36  xyz. 

21.  1,  -  2,  3,  -  4,  5,  -  6. 

22.  —  3  a,  7  a,  42  x,  —  13  a. 

23.  What  are  the  units  of  addition  in  examples  1  to  8  ? 

24.  (—  5a)  +  7a+(—  12a)=—  10a.  '    Is    this    true   when 
a  =  1  ?     when  a  =  2  ?     For  what  other  values  of  a  is  it  true  ? 

25.  In  what  respect  may  two  like  terms  differ  ? 

26.  What  kind  of  algebraic  expression  is  obtained  by  add- 
ing two  like  monomials  ? 

27.  Solve  2  a +9  a +  (-3  a)  =  17  -f-(-l). 

28.  Evaluate  a  -f-  b  +  c  +  d,  when  a  =  2,   b  =  —  3,  c  =  —  5, 
d=-5. 

29.  Simplify  a  +  6 4- c 4- a"  when  a  =  —  3a,  &=— 5a,  c  =  a, 
d  =  4a. 


Addition  of  Like  Monomials  39 

Solve  the  following  equations : 

30.  2.5  x  +  12  =  312. 

31.  18a>  +  (-12a>)=15+(-12). 

32.  6a?+(-5s)=1.7  +  3.3. 

33.  2x  +  (—  3x)  +  4:X  =  —  5.1. 

34.  18aj+(-12»)4-20a;  =  164-(-3). 

35.  1.25z4-(-.75x)=.95+(-.45). 

36.  24  y  +  (- 12  y)+  Sy  =  125  +(-  32)  +  (-  86)  +  48. 

37.  3aj  +  12aj  +  48aj+(-50a;)  =  48+2(-4)  +  (-l). 

38.  _7i,-f-10p+(-jp)=2  x  (-3)  +  (-3)«. 

39.  _l0r  +  (-8r)  +  (-7)  +  T  =  10(-2)  +  (-5)2. 

73.  In  arithmetic  we  can  add  several  numbers  in  any 
order  and  get  the  same  sum,  and  we  can  multiply  several  factors 
together  in  any  order  and  get  the  same  product.  Likewise  in 
algebra  we  can  rearrange  the  terms  of  a  polynomial,  or  change 
the  order  of  the  factors  of  a  term  without  changing  its  value. 

Thus,  5-2a  +  3&=3&-2a  +  5  =  3o  +  5-2a.  Also  6  m*np 
—  6  nm2p. 

74.  Arranging  the  Terms  of  a  Polynomial.  It  is  generally 
convenient,  and  sometimes  necessary,  to  arrange  the  terms  of 
a  polynomial  in  some  particular  order.  The  most  frequent 
arrangement  is  in  descending  powers  of  some  letter  that  occurs 
in  all,  or  all  but  one,  of  the  terms. 

Thus,  2  x:i  +  3  x2  -  2  x  +  8,  4  x4  -  3  x  -  7,  ax*  +  bx  +  c  are  all  ar- 
ranged in  descending  powers  of  x. 

An  arrangement   in   ascending  powers   is  sometimes   used. 

Thus,  — 7  —  3  a;  +  4  x4  represents  one  of  the  above  expressions  re- 
arranged in  ascending  powers  of  x. 

If,  instead  of  different  powers  of  the  same  letter,-  we  have 
the  same  power  of  different  letters,  we  generally  arrange  the 
terms  alphabetically. 

Thus,  a2  +  b2  +  c2  is  arranged  alphabetically. 


40  Addition 

ORAL  EXERCISE 

75.  Rearrange  the  following  (a)  in  descending  powers,  (b)  in 
ascending  powers  : 

1.  tf-l+2x*-lx.  4.   &  +  16b-llb*  +  5V-l. 

2.  3  a +  7  -x*.  5.   p-2p2  +  7. 

3.  3  m2  —  4  m  +  6  -f  m3.  6.    bx  -f  a#2  +  c. 

Rearrange  the  following  alphabetically : 

7.  262-a2  +  3d2-5c2.  io.   ft2 -  14/2  +  16 #2  +  7 &2. 

8.  c—  b  —  a  —  d.  11.   y  —  x  +  z  —  v  +  w. 

9.  n2  +  2m2-l2.  12.  p3  -  r3  +  q3  +  n\ 

ADDITION  OF  UNLIKE  MONOMIALS 

76.  Just  as  in  arithmetic  2  ft.  -f-  3  ft.  =  5  ft.,  so  in  algebra 
2a+3a  =  5a;  but  just  as  2  ft.  and  3  in.  cannot  be  added 
without  changing  them  to  the  same  denomination  (2  ft.  +  3  in. 

•  =  24  in.  +  3  in.  =  27  in.),  so  the  sum  of  2  a  and  3  b  must  be 
expressed  in  the  form  2  a  -f  3  b  until  we  have  the  values  of  a 
and  b. 

ORAL  EXERCISE 

77.  1.  What  is  the  length  of  a  fence  around  a  field  40  rods 
long  and  30  rods  wide  ?  40  rods  long  and  x  rods  wide  ?  a  rods 
long  and  b  rods  wide  ? 

2.  A  ship  travels  300  miles  one  day  and  320  miles  the  next 
day.  How  far  has  it  gone  ?  If  the  number  of  miles  had  been 
a  and  6,  how  far  would  it  have  gone?  What  is  the  value 
of  this  result  if  a  =  300,  b  =  320? 

3.  There  were  x  boys  and  y  girls  in  school  last  term.  If  a 
new  boys  and  b  new  girls  enter  this  term  and  m  boys  and  n 
girls  leave,  how  many  boys  and  how  many  girls  are  there  in 
school  ?  how  many  boys  and  girls  together  ? 


Addition  of  Unlike  Monomials  41 

78.  The  following  statement  may  be  regarded  both  as  a 
definition  of  the  sum  of  unlike  terms  and  as  a  rule  for  adding 
such  terms : 

The  sum  of  several  unlike  terms  is  the  algebraic  expression  obtained 
by  uniting  them  with  their  respective  signs. 

Thus,  the  sum  of  2  a,  (—  3  6),  and  11  c  is  2  a  —  3  b  +  11  c. 

79.  It  is  often  necessary  to  use  the  rule  for  adding  like 
monomials,  along  with  the  above  rule  for  adding  unlike  mono- 
mials. 

Thus,  Sa  +  5b+(-$b)  +  (-2a)=a-3b. 

This  kind  of  simplification  is  usually  called  collecting  terms. 

ORAL  EXERCISE 

80.  Add  and  arrange  the  terms  in  proper  order : 

1.  -  a2,  -  4a2,  2a2.  5.  |a2,  -  §  62,  Ja2,  \b\ 

2.  3  6,  -2  a,  a,  -5  b.  6.  4  m2, -5  n2,. 5  m2, 4  n2, 2  m2. 

3.  4a2,  -  3x,  7,  5x,  -2x\  7.  3x\  -  x,  2,  4a2,  -  1,  x,  3. 

4.  -1,  2x,  3x?,  -x%  1.  8.  46,  -2  a,  -26,  4c,  -  a; 

Collect  the  terms  and  arrange  in  order: 

9.    46+(-7)+36  +  26+(-36). 

10.  8x  +  y  +  2z  +  3x  +  y  +  7z. 

11.  7y  +  x+(-2x)+2y+(-z). 

12.  7*+(- </)  +  (- 2a)  +  (-z). 

13.  a  +  26+3c  +  2a+(-76)  +  (-9c). 

14.  l4-m  +  w4-2?  +  n+(-2m)4-(-4 

15.  5E  +  7E=?     5-15  +  7.15  =  ? 

16.  3-a  +  9-a  =  ?     3.7  +  9-7  =  ? 

17.  ia  +  3ia  =  ?        1. 17  +  31-17  =  ? 

18.  17a  +  3a  =  ?       17-13  +  3-13=? 

19.  13-21+(-5-21)  +  (-7.21)=? 


42  Addition 

EXERCISE 

81.  Collect  terms  and  arrange  the  results  in  proper  order: 

1.  3a+(-5c)  +  (-a)+36+(-  c). 

2.  -5a+  (+l.la?)+  .7  a2  4- (- 2.3  a;2). 

3.  22m  +  6n+(-14/))+2m+(— 10n)  +  4p+(-2m) 
+  10n.'4-18j>. 

4.  Ja4-|6+(-a)+fc+(-f5)+ia+(-fc). 

s-  i»+(-*y)+f»-K-"iiO+(-«). 

6.  56a  4  58 p  +  218  4-  92  p  +  36a  +  74  4-  20p  +  360. 

7.  54m  +  (-62n)  +  18»  +  (-62m)  +  (-6a;)+42w  +  10m 
+  18  n'+  (—14®). 

8.  10  m  +  11  +  (-  5  a?)  +  (-  12)  +  (-  4  m)  +  (-  3  a?)  +  1 

+  9x4-  (—5  m). 

9.  13»  +  (-5y)  +  8aJ4-(-5aj)4-9y  +  (-lla;)+(-3a;)  I 
+  (-6y)+ft. 

10.  5.67)  +  18.5  g  -f  (-  7.25 jp)  +  11.5  q  4- 15.5  p  +  (-  9.4  q). 

11.  5a+(-3!&)  +  5fc4(-6Ja)49ii&43fa4(-2jic). 

ADDITION  OF  POLYNOMIALS 

82.  1.   Add  and  compare  : 

2  ft. +  3  in.  2/4-    Si 

4  ft. +  2  in.  4/4-    2i 

7  ft.  4  5  in.  7/+    5i 


13  ft. +  10  in.  13/ 4  10 1 

2.   Add  2a-36  +  7c,26  +  a-2c,  c4-2a-36. 
2a-3&4-7c 

a  4-  2  b  —  2  c  Rearrange  so  as  to  have  like  terms 

2  a  —  3  b  4-     c         in  the  same  column  ;  then  add. 

5a-464-6c 


Addition  of  Polynomials  43 

3.-  Add  2  a2  -3a  +  7,  -4  a2  -6  +  5  a,  8  a2  -9a-  7. 
2a2 -3a +  7 
-4a2  +  5a-6 
8a2 -9a— 7 


6a2 

-7a- 

6 

83.  .4c7a*: 

ORAL 

EXERCISE 

1.   3a +  26       4. 

9x 

-9 

7. 

4a6  +  c       10.   8 

z  +  9 

4a  +  96 

8x 
5r 

-3 

+  3 

8.    2 

-  5ab  —  c 

3 

2.   4  a  +3          5. 

a +  3              11. 

a; 

7#  +  8 

r 

-2 

5 

a -3 

■a;  +  3 

3.   5m +  3         6. 

r 

+  2s 

9.    4p  +  7              12. 

a;  —  5 

8m-2 

3r 

-5s 

5: 

P                             - 

£  +  5 

13.      a+     b-    c 

20. 

4r  +  2s—     t 

2a-2b-  3c 

-3*  — 2* 

14.   3a -6  + 2c 

2r  +  7s 

a         —    c 

21. 

iC2  +  #  —  1 
—  X2  +  iC  —  1 

a2  -  X'  + 1 

15.   4a  —  b  —  2c 

6  +  2c 

16.           a2 -2a +  5 

22. 

3a-    2/  +  2z 

-2a2  +  3a- 

2 

—    x  +  Sy—'Sz 

17.    2^  +  3.^  +  9 

-2x-  2y  +  2z 

x2            -9 

23. 

a  +  6  —  c 

a  —  b  +  c 

— a+6+c 

18.   a2  +  a 

2a  +  l 

19.      a  -  3  6 

24. 

a2  -  2  a  +  1 

2a            -    c 

a*  +  2a  +  l 

36  +  2c 

-2a2            -1 

44  Addition 

84.  Checking  Results.  2  a  +  3  a  =  5  a  for  all  values  of  a.  The 
sum  of  3  a  —  5  b  and  4  a  +  2  6  equals  7  a  —  3  6  for  all  values  of 
a  and  b.     This  fact  may  be  used  to  check  the  answers. 

1.  Add  3  a—  5  b  and  4  a  +  2  6  and  check  the  result. 

Addition  Check.     a=b  =  1. 

3a-5b  3-5=-2 

4a+2fr  4+2=      6 

7a-36  7-3=     4 

The  work  on  the  right  is  the  result  of  putting  a  =  1  and 

6  =  1  in  the  two  expressions  to  be  added  and  in  the  result. 

Notice  that  the  final  number,  4,  is  the  algebraic  sum  both  of  the 

last  line  and  the  right-hand  column ;   that  is,  7  —  3  =  4  and 

-2  +  6  =  4. 

2.  Add  and  check  the  result,  2  a2— 3 a +4,  7  a+3,  -2  a2+5. 

Addition  Check,     a  =  2. 

2a2-3a+4  8-6+4=        6 

7a+3  14+3=      17 

-2a2            +5  -8         +    5=-    3 

4  a  +  12  8  +  12  =      20 

85.  To  add  two  or  more  polynomials  : 

1.  Arrange  the  terms  in  order  of  powers  of  some  letter  (or  alphabeti- 
cally), writing  like  terms  in  the  same  column. 

2.  Add  the  like  terms  in  each  column  and  unite  the  partial  results 
obtained  with  their  respective  signs. 

Note.  Check,  or  prove  the  correctness  of  the  result,  by  substituting  a 
numerical  value  for  the  letters  used. 

Examples  t 
1.   Add  and  check,  3  a  -  2,  a2  -  1,  2  a  +  7  a2  +  3,  a  -  5. 
Addition  Check,     a  =  1. 

3a-2  3-2=        1 

a2  -1  1-1=0 

7a2  +  2a  +  3  7  +  2  +  3=      12 

a-5  1-5=-    4 

8a2  +  6  a-  5  8  +  6-5=       9 


Addition  of  Polynomials  45 

2.   Add  and  check,  3  a2-  2  ab  +  b2,  -  4  aft  +2  62  -  a2,  a2+62. 

Addition  Check,     a  =  6  =  1. 

3a2_2a&+     ft2  2 

-     a2-4a&  +  262  -3 

a2  +     &2  2 

3a2-6a6  +  462  1 

EXERCISE 

86.   Add  the  following,  rearranging  when  necessary,  and  check 
the  results  in  examples  1  to  7 : 

1.        3a +  26  4.        lab  —  3ac  +  46c 

4a  —  76  4  a6  +  4  ac  —  5  6c 

—  5  a  -f  4  6  —  5  a6  —     ac  -+-  26c 

5. 


6. 


7.  3a2  +  7a6-262,   962-3a6  +  a2,   5a2  +  762. 

8.  14a-66  +  3c-5d,  9a  +  7  6  -  4c  -  9d\ 

9.  .8  a2  -  3.47  a6  -  17.25  ac  +  3.75  6c, 

-  7.5  a2  +  .47  a6  +  12  ac  -  7  6c. 

10.  1.5  x2  -  3.2  x  -  .07,  8.04  x  -  2.1  +  4  z2,  .3  a  -  .75  <c2. 

11.  14a-66  +  3c-5d,  9a  +  7  6  -  4c- 9a\ 

5a-6  +  c  +  14a\ 

12.  3  a2  -  7  62  +  10  c2,  62  -  7  c2  +  3  a2,  21  c2  -  7  62  +  a2. 

13.  27  a6  +  5  ac  +  6c,  —  4  ac  —  21  a6,  a6  +  43  ac. 

14.  a2  +  x  -  10  +  2  ax,  2  a2  -  3  x  +  20  4-  ««, 

-5a2-3z  +  30  +  5ao;. 

15.  5  a26  -  7  a36c  -  13  62c4  +  10,  12  a26  +  8  a36c  -  10  62c4  +  2. 


2. 

±x2  +  7a_3 

5aj»-7a?  +  2 
9a;2-5a;  +  l 

3. 

2a2-7a6  + 
4  a2  +  7  a6 
-3  a2              - 

62 

7  62 

3a2+   4a 

-    7 

-10a2 -12a 

4-11 

7a2  +    8a 

-   4 

5  vx  —    6  v 

+  4sc 

—  11  va  +  13  v 

-7a 

vx  —       -y 

4-    a 

46  Addition 

Add  the  following : 

16.  5  m2  —  6  it2,  —  3  m2  +  4  mn  4-  5  7i2,   —  m2  -  3  ran,  -f-  2  n*. 

17.  7a-36  +  5c-10a',  26-3c  +  d-4e, 
5c-6a-4e  +  2a",  -3  6-8  c  +  7a-  e,  21e  -  16c  +  a  -  5a\ 

18.  3a62-4a26+a3,   -4ac2+  5a62-  c3,  -763+2a26-6ac2, 

5  a3  -  11  a62-  12  ac2. 

19.  a3  +  3a26  +  3a62  +  63,  -  5a62  +  3a26  -  63  +  3a3, 

3a62-5a26,  363-3a3,  -  563  +  2a26  -  4a3  +  3a62. 

20.  5a;2-h9,  3cc-5,  4ic2-5»-l,  -  Sx2  +  2a;  +  6. 

21.  p2g  —  a3  -f-  j)3  —  pa2,   a3  —  pq2  -f  p2g  —  pz,  p3  +pq2  —  p2o. 
In  examples  22  to  31,  X  =  2a2  +  3  a  -  1,  F=-3a3  +  4a, 

Z=3a3-5a2  +  2a,    P=  a3  -  a2  -  a  -  1,    Q  =  15a2  +  a-3, 
R  =  —  a3  +  7a  —  5. 

22.  FindX+F+Z.  25.   Find   P+Q+X+F 

23.  Find  1+7+7-^5,  26.   Find  X  +  F+  72  +  ft 

24.  Find  X  +  Y  +  R.  27.    Find  R  +  P  +  Z  +  X. 

28.  Find  X+  Q +(- 13a2  -  4a  +  4). 

29.  Find  X+  F+Z  +  (3a2-5a  +  3). 

30.  Find  P+Q+R  +(14  a2  -  8  a  -  10). 

31.  FindX+  Y+Z  +  P+Q+R. 

REVIEW  EXERCISE 

87.   The  four  kinds  of  algebraic  addition  are : 

1.  Addition  of  positive  and  negative  numbers  without  literal  parts. 

2.  Addition  of  like  monomials. 

3.  Addition  of  unlike  monomials. 

4.  Addition  of  polynomials. 

1.  State  the  rule  for  each  of  the  four  kinds  of  addition. 

Add  the  following : 

2.  5  a,  (-3  a),  7  a,  (-32  a). 

3.  -7a,  (-46),  (-196),  226,  8a. 

4.  3  x,  3  y,  —  3  x,  —  2  y,  x. 


Review  Exercise  47 

5.  3a2  +  262,  7a2-862,  462,  2a2  +  76*. 

6.  -3a2  +  7a-2,  4a-7,  2a2  +  9,  a^-a-l. 

7.  ±a-\b  +  \c,  2c-3d  +  a,  £6-fa  +  a\ 

8.  2  a  +  7  6  —  3  c,  -56  —  4a,  —  6  —  c. 

9.  1.5  m2  —  f  mn  —  n2  +  (  —  .5  m2)  +  f  mn  +  3.5  n2. 

10.  6  a6  -  a2  +  62  -  5  ab  +  2  62  -  3  a2  +  2  a6  +  6  a2. 

11.  2.5  xy  -  3.5  ay2  +  4.5  x2y  -  1.25  ay  +  2  an/2  -  2  x2y. 

12.  |)g2  +  p2o  +  p2,  q2  +  p2q  +  3#2,  4pa2  -  3  q2  -  2p2. 

In  examples  13  to  17A  =  x2—3x  +  7,  B  =  ox2  +  7  x  —  7, 
C  =  a2-6,  D  =  2x2-^x-3. 

13.  Find  .4  +  5+ (-2  a2). 

14.  Find  C  +  B  +  D. 

15.  Find£  +  Z)+C  +  (-4a2). 

16.  Find  A  +  C  +  B. 

17.  Find  A  +  C  +  D. 

18.  Determine  by  substituting  numerical  values  whether 
a  +  6  —  2  c  is  the  sum  of  3a  +  26  —  5  c,  —  4a  +  76  +  c,  and 
2a- 86 -f  3c. 

19.  Add  3  x2  +  1,  x  +  2,  4  a2  +  x  +  3,  and  check  by  putting 
a  =  2. 

20.  Collect  the  following  terms  :  13a+56  +  (— 3a)  +  (-7  6) 
+  8a+(-3). 

Add  the  following  : 

21.  3a,  a  +  7  6  —  4c,  —  3a  —  56  +  3 c,  —  6  +  c. 

22.  -2a  +  5a2  +  7,  3a  +  5-6a2,  4a2  +  4a-3. 

23.  7  a  -  6  6  +  8  c  +  3  d,  13  a  -  15  6  -  7  c  -  11  d, 

-6o*  +  56  +  7c-2a,  _  5  c  +  10  d  +  28  6  -  17  a. 

24.  234  a  +  36  y  +  18  z,  24  y+12«  -  12*,  25  y  -  44  a  -  16  2, 
12a -17z,  -85t/  +  6a. 

25.  5n2-7ps-8m2,  -2p2  +  m2  +  3n2,  9m2  -  8n2  +  7p2, 
-2m2  +  2p*. 


48  Addition 

Add  the  following  ? 

26.  5.2  x  +  .05 2/  — 2.1  z,  - .6  a; - .5  y  +.1  2,  3.5  x  +  .7  ?y-  .5 2. 

27.  2  a2  -  5  +  8  x,  2  -  4  x,  17  -  x  -  x\ 

28.  3  a -^7  6,    -8c  +  4d-8e,   7a  +  6e  +  9c-5d  +  8  6. 

29.  1.34  m  -  7.6  71  -  .397  j>,  -  81.7p  -  9.4  m  -  8.7  n, 

9.76  ra  +  4.33p  +  9.3  n. 

30.  41.6  q  -  43.1  a  +  37.8  y,  .09  ?/  -  5.37  x  -  4.05  g, 

1.97  x  -  4.1  ?/  -  .8  7. 

31.  .3  x2  +  .1  #2  -  .3  yz  -  .1  z2,  .2  a#  -  .3  #2  -f-  .3  yz, 

-  .4  x1  -  .2xy  +  .1  y2  +  .1  z\ 

32.  Solve  3  x  +  2  x  +  5  +  9  =  27  +(-3). 

33.  Solve  1  x  +(_  3  #)  +  (-»)=  5+ (-3)+ 12. 

34.  Find  the  sum  of  five  numbers,  the  first  number  being 
2  x  and  each  succeeding  number  being  3  a  greater  than  the 
preceding. 

35.  If  a  passenger  ticket  costs  x  cents  a  mile  and  it  costs 
4  cents  to  carry  a  bicycle  each  25  miles,  how  much  is  the  cost 
of  both  for  250  miles  ? 

36.  Through  how  many  degrees  of  longitude  does  a  ship  sail 
in  going  from  -  18°  to  +  37°  ? 

37.  The  oldest  known  mathematical  manuscript  was  written 
about  —  1700  (1700  b.c).     How  long  ago  was  it  written  ? 

38.  A  merchant's  capital  was  diminished  by  $  1400  and  then 
amounted  to  $  4500.    What  was  his  capital  at  first  ? 

Suggestion.     Let  x  =  number  of  dollars  at  first. 

39.  At  a  certain  election  A  received  113  more  votes  than  B. 
The  number  of  votes  cast  for  both  was  847.  How  many  votes 
did  each  receive  ? 

Suggestion.  Let  x  —  number  of  votes  B  received. 

Hence  x  -f  113  =  number  of  votes  A  received. 
Then  x  +  x  +  113  =  847.     (By  the  conditions.) 


Review  Exercise  49 

40.  A  rectangular  field  is  twice  as  long  as  it  is  wide  and  its 
perimeter  is  360  rods.   Find  the  leng'th  and  the  width  of  the  field. 

41.  A  ball  team  played  20  games  and  won  three  times  as 
many  as  it  lost.  How  many  games  were  won  and  how  many 
were  lost  ? 

Suggestion.    Let  x  =  number  of  games  lost. 

42.  A  boy  paid  x  cents  for  a  bat,  twice  as  much  for  a  ball, 
and  20  cents  less  for  a  mask  than  for  both  ball  and  bat.  How 
much  did  each  cost  him  if  he  spent  $  2.20  all  together  ? 

Suggestion.     Change  $  2.20  to  220  cents. 

43.  The  larger  of  two  numbers  is  three  times  the  smaller  and 
their  sum  is  84.     Find  the  numbers. 

44.  The  larger  of  two  numbers  exceeds  the  smaller  by  10, 
and  the  sum  of  the  two  numbers  is  94.     Find  the  numbers. 

45.  The  girls  in  a  certain  high  school  outnumbered  the  boys 
by  122.  The  entire  enrollment  was  2742.  How  many  boys 
were  there  in  the  school? 

46.  A  woodworking  class  spent  $  32.50  more  for  jack  planes 
than  for  try-squares.  If  both  tools  together  cost  $  50,  find  the 
cost  of  each  kind. 

47.  One  farmer  by  spraying  his  potatoes  raises  30  bushels 
more  on  an  acre  than  his  neighbor.  If  both  together  raise  400 
bushels,  how  many  bushels  does  each  raise  ? 

48.  In  1910  Jerry  Moore  of  South  Carolina  won  a  prize  in  a 
boys'  corn  raising  contest.  In  1913  Walker  Dunson  of  Alabama 
raised  4  bushels  more  corn  on  an  acre  than  Jerry  Moore's  record 
yield.  The  total  yield  on  the  two  acres  was  460  bushels.  How 
many  bushels  did  each  raise  ? 

49.  In  1914  the  Allred  boys,  Luther,  Clarence,  Elmer,  and 
Arthur,  of  Georgia,  raised  on  four  one-acre  plots  of  land  824 
bushels  of  corn.  Clarence  raised  10  bushels  more  than  Elmer 
and  7  bushels  less  than  Luther,  while  Arthur  raised  43  bushels 
less  than  Elmer.     How  many  bushels  did  each  raise  ? 


IV.    SUBTRACTION 

SUBTRACTION  OF  LIKE  MONOMIALS 

ORAL  EXERCISE 

88.    1.   Define  subtraction.     (§  42.) 

2.    State  the  rule  for  subtracting  signed  numbers.     (§  44.) 

Subtract  the  following : 

3.  3-4;  7-8;  10-15. 

4.  3-(-4);  9 -(-15);  4 -(-4). 

5.  -3-5;  -7-8;   -15-2. 

6.  _1_(_3);  -3-(-4);  -5-(-8). 

7.  7  ft. -5  ft.;  7f-5f. 

8.  20°-30°;  20d-30d. 

9.  601b. -351b.;  60  p- 35  p. 

10.  85  acres  —  42  acres ;  85  a  —  42  a, 

11.  $73-  $21;  73  d  -21c?. 


12. 

-17°-5°;  -17  d-5d. 

13. 

—  10  #  —  3  <c ;  —  5  a  —  3  a. 

14. 

-17  a; -22  a;    -  8  x  -  8  x. 

15. 

20 -(-10);  20  c?-(- 10  cT). 

16. 

8_(_5);  Sx-(-5x). 

17. 

18  -(-1);  lSr-(-r). 

18. 

49xy-4:5xy;  49  •  12-45  •  12. 

89.   From  the  examples  of  §  88  we  derive  the  following  rule : 

To  subtract  a  monomial  from  a  like  monomial,  change  the  sign  of  the  sub- 
trahend and  add  the  resulting  number  to  the  minuend.     (See  §§  42  to  44.) 

The  student  should  change  the  sign  mentally. 

60 


Subtraction  of  Like  Monomials 


51 


Examples 
1.    Subtract  11m  from  17  m.      2.    Subtract  4  x  from 


Add  —  11  m  to  17  m. 


17  m 
11m 
6m 
Check.     Add  6  m  to  11  m. 


Add 


6x. 
4  a:  to  —  6  x. 


-6x 

4ff 

-10# 
Check.     Add  —  10  x  to  4  #. 


3. 


9a  —  4a  =  —  13  a. 


The  minuend  is  —  9  a,  the  subtrahend  is  4  a.     Therefore  add  —  4  a  to 
—  9  a.     Let  the  student  check  the  result. 

4.  —  5?*  —  (— 7r)  =  2r. 

Add  -f  7  r  to  —  5r.     Let  the  student  check  the  result. 

5.  6a-8a=-2a.     Why? 

EXERCISE 

90.    Subtract  the  following : 


1. 

15  a 

5. 

2a 

9. 

15  mn* 

13. 

—  m«p2 

2a 

a 

17  mn2 

mnp2 

2. 

21  & 

6. 

11m2 

10. 

0 

14. 

—  mnp2 

23  6 

7  m2 

-5 

—  mnp2 

3. 

7a 

7. 

Ixy 

11. 

0 

15. 

5m3n2 

10a? 

-3xy 

—  5a 

-  18  mhi2 

4. 

-7x 

8. 

-  11  ab 

12. 

—  21  xyz 

16. 

-18a>Y 

10  x 

-    2ab 

—    4  xyz 

—  27  ary 

17.  —5  be  from  —  3  6c. 

18.  18  65  from  -4  65. 

19.  -  13  a26  from  24  a26. 


20.  3  a62  from  -  7  a62. 

21.  —5cd?  from  14  cd?. 

22.  —  2.25  m^n  from  —  3.5  m2n. 

23.    From  the  sum  of  7  a62c  and  —  11  a62c  take  —  4  a62c. 

24    Take  the  sum  of  whi  and  —  6  m2n  from  the   sum  of 
—  4  mfri  and  3  m2?i. 


52  Subtraction 

25.  The  minuend  is  0  and  the  subtrahend  is  —3  a;.  What 
is  the  difference  ? 

26.  The  minuend  is  —  27  xy  and  the  difference  is  5  xy. 
What  is  the  subtrahend  ? 

27.  The  subtrahend  is  —  5.2  x  and  the  difference  is  .05  x. 
What  is  the  minuend  ? 

SUBTRACTION  OF  UNLIKE  MONOMIALS 
ORAL  EXERCISE 

91.  1.   What  are  unlike  monomials  ? 

2.  What  does  a  —  b  mean  ? 

3.  What  length  remains  if  10  feet  are  cut  from  a  rope  32 
feet  long  ?  if  x  feet  are  cut  from  a  rope  32  feet  long  ?  if  b  feet 
are  cut  from  a  rope  a  feet  long  ? 

4.  How  much  have  you  left  if  you  have  16  cents  and  spend 
7  cents  ?  if  you  have  16  cents  and  spend  x  cents  ?  if  you  have 
a  cents  and  spend  x  cents  ? 

5.  If  you  throw  a  stone  vertically  upward  ft  feet,  how  high 
is  it  after  it  has  fallen  d  feet?  How  high  is  the  stone  if 
h  =  62  and  d  =  21  ? 

6.  If  the  enrollment  in  a  class  is  m  girls  and  n  boys  and 
there  are  x  girls  and  y  boys  absent,  what  is  the  attendance  ? 

7.  How  are  unlike  monomials  added  ? 

92.  To  subtract  a  monomial  from  an  unlike  monomial,  change  the 
sign  of  the  subtrahend  and  add  the  resulting  number  to  the  minuend. 

Examples 
1.   Subtract  2  a  from  3  x. 

Sx 

9  The  subtrahend  when  its  sign  is  changed  becomes  —2a; 

adding  this  to  3  x  gives  3  x  —  2  a. 

3x  —  2a 

The  result  may  be  checked  as  usual,  3«  —  2a  +  2a  =  3a;. 


Subtraction  of  Unlike  Monomials  •  53 

2.   Subtract  —  6  from  a. 


a 

fi             The  subtrahend  with  its 

we  have  a  -+-  6. 

a  +  6 

i  sign  changed  becomes  6.     Adding, 

3.   Subtract  —  3  from  4  x. 

4:X 

_3            4z-(-3)=4a;  +  3. 

4x  +  3 

EXERCISE 
93.   Subtract: 

1.    116  from  17  a. 

3.    —5b  from  2 a. 

2.    —  7  a  from  5  ?/. 

4.    b  from  —  6  a. 

5.    7x                6.   5a26 

7.         ?     '          8.       d 

-7                     3a&2 

—  p                    —  c 
i 

9.   7  c2  from  5ab. 

11.    —  5  a?/2z  from  —  kxhjz. 

10.    —  4  ?/  from  3  x2. 

12.    —  a  from  a  +  2b  +  a—  2b. 

13.  -  10  from  3  x  +  2  #  +  2/-  3  a?. 

14.  10  from  3a +  2*/  +  (—  2?/)  —  7 x. 

Collect  terms: 

15.  3a  +  5?/-5z-2a  +  (-3z)-3z-2a. 

16.  -4  +  a-(-a)-f8. 

17.  4a-5a+(-2a)+76-(-36). 

18.  _5+(-7)-3-8-(-17)-(-6). 

19    12  a  +  3a  —  4z—  4a  —  (—  5z)-z  +  x  —  6  a. 

20.  3a  +  4?/-(-5z)-2a-3?/  —  4z -(- a)+ # -(- z). 

21.  22a -23?/ -(-24a). 

22.  44  +  21  z  -(-22). 

23.  17a-(-lla)  +  (-13  6)-16&. 

24.  21c-15c  +  28d-(-6cT). 

25.  18a-21a-10d  +  8d. 


54  Subtraction 

SUBTRACTION  OF  POLYNOMIALS 
ORAL  EXERCISE 

94.   Subtract  the  following  : 

1.  111b.  7  oz.     11^  +  73  4.    12  mi.  20 rd.    12m+20r 

8  lb.  5  oz.       8p  +  5z  8  mi.  8m 

2.  7  ft.  9  in.          7/+  9  i           5.   3  ft.  7  in.          3/+  7  i 
4  ft.  3  in.         4/  +  3 i  5  in.  5i 

3.  13 mi. 40 rd.    13m+40r        6.    5a  +  2b  8x-5y 
11  mi.  28  rd.    llm+28?-              2g  +  46  7a;  +  4y 

3a-2b  x-9y 

Let  the  student  check  the  results  in  the  last  example  by  adding  the 
difference  to  the  subtrahend. 

7.   7a;2-3a;  11.   3a  +  26  +  7c 

4  a;2  —  5x  a  +     b  +    c 


2a6  +  7c 
8  ab  +  9  c 

17  6  +  2p 
146  +  3p 

—  5x  +  7y 
-Sx-Sy 

12. 

5a-86  +  7c 
2a  +  36-4c 

13. 

4a;  +  3?/  —  7z 
5x-2y-Sz 

14. 

6x —y — z 

2x-\-y  —  z 

10. 


95.   From  these  examples  we  derive  the  following  rule : 

To  subtract  one  polynomial  from  another,  write  the  subtrahend  under 
the  minuend,  with  like  terms  in  the  same  vertical  column.  Change  the 
sign  of  each  term  in  the  subtrahend,  and  proceed  as  in  addition. 

Examples 
1.   Subtract  6a;2  -  3a;  -  12  from  15  x2  +  8 x  + 1. 
Subtraction        Check,    x  =  1. 

15  a? +  8a;+  1=  24  This  result  might  be  checked  by  adding  the 
6  a;2—  3  a;— 12=—  9  difference  to  the  subtrahend.  The  sum  should 
9  a;2  +  lla;-f-13=     33    equal  the  minuend. 


Subtraction  of  Polynomials  55 

2.  From  6  mn+3  m2  +  n2  take  5  n2  +  ra2  —  3  ran. 

3m2  +  6mn+    n2 

,      o  ,    cr    o        Arrange  the  terms  in  descending  powers  of  m. 

ra2  —  3ra?i  +  5n2        _       &      ...  ,.     ,.„  "•*       _.    ,      , 

— -        Check.     Add  the  difference  to  the  subtrahend. 

2  m2  +  9  ra?i  —  4  n2 

3.  Subtract  4  a  —  3  b  -f  c  from  2  b  — 3  c. 

2b-3c 
4  a  —  3  b  -f-     c        Let  the  student  check  the  result. 
—  4a  +  5  6  —  4c 

EXERCISE 
96.   Subtract,  and  check  results  as  directed : 

2.    8.1  r-  1.5  s       .      3.   5  a  +  b 
4    r+    .2s  8a 


1. 

13  ra  +  40  r            : 
llra  +  89r 

4. 
5. 

3a 
2a-6 

3a  —  b—    c 

2a        +5c 

6. 

a  —  2b  +  c 
5a-56 

7. 

2.5r-4s  +  3 
1.3r-2s 

8. 

8a 

6a-6  +  3c 

9. 

21d  +  14ra  +  27s 
16d  +  18ra  +  27s 

10. 

11  h  +  41m  +  56s 

7  h  +  59  m  +  34  s 

11. 

36ra  +  37r  +  42?/ 

25ra  +  71r  +  84?/ 

12. 

2a-36 

5a            -7 

13. 

a;2  +  3a;_7 

5  x2  —     a;  —  1 

14. 

2a+     6 

a  —  36  +  7c 

15. 

56        -2 

—  4a           —  c 

16.  Subtract  2  a  —  3  b  +  c  from  a  +  b  +  c. 

17.  4  m2  —  5  mn  —  7  n2  minus  3  n2  +  ra2  —  5  mn. 

18.  The  subtrahend  is  —  2,  the  minuend  is  a  -f  b  —  2.    Find 
the  difference. 

19.  The   difference   is    a  +  b  —  c    and    the    subtrahend   is 
2  a  —  3  6  +  c.     Find  the  minuend. 


56  Subtraction 

20.  From  the  sum  of  4  a2  +  3  ab  +  7  b2  and  b2  —  7  a2  take 
a2  +  62  -  a&. 

Collect  terms  in  examples  21  fo  25. 

21.  (_7)  +  3-(-5)+12+(-12)-8. 

22.  5a  +  2&-36+(-26)-(-7a). 

23.  —  8ra+(—  3n)+4ra  +  6n  —  7ra+n. 

24.  10+(-14)-6  +  ll+(-4)-(-13).         < 

25.  7i>  +  8g-(-3j9)  +  (-7g)+4p-llg. 

26.  From  4  a?2  -f  2  <ey  —  3  y2  take  »2  —  xy  +  2  ?/2. 

27.  From  a3  +  3.a26  +  3  «62  +  63  take  a3  -  3  a2b  +  3  a&2  -  b\ 

28.  From  the  sum  of  m2  +  3  mn  —  n2  and  2  m2  —  5  mn  +  w2 
subtract  2  m2  —  2  mn  +  n2. 

29.  Subtract  the  sum  of  ax2  +  bx-\-  c  and  2  asc2  —  3  6a?  —  2  c 
from  4  aa?2  —  2  fo»  +  c. 

30.  What  must  be  subtracted  from  a+b-\-c  to  give  a+6— c? 

31.  What  number  added  to  3ax  +  &by  —  7  cz  will  give 
ax  -h  by  4-  cz  ? 

32.  The  sum  of  two  algebraic  expressions  is  3  a3  —  4  and 
one  of  them  is  cc3  +  x2  +  1.     What  is  the  other  ? 

33.  If  dates  b.c.  are  considered  negative  and  a.d.  positive, 
how  many  years  are  there  from  —  509  to  —  27  ?  from  —  34 
to  +  48  ?     from  -  480  to  +  60  ? 

34.  From  the  sum  of  2  a2  —  3  ab  +  4  ft2,  a2  +  2  ab  —  2  62,  and 
2  a&  subtract  4  a2  —  a&  —  2  62  plus  a2  +  ab  +  &2. 

35.  From  a2x*  -f-  a&a?3  +  bcx2  subtract  a2x*  —  aba?  —  bcx2. 

36.  Write  five  consecutive  numbers  of  which  x  is  (1)  the 
largest,  (2)  the  smallest,  (3)  the  middle  number.  Add  the 
5  numbers  in  each  of  the  parts  (1),  (2),  (3).  Which  gives 
the  simplest  result? 


Parentheses  57 

37.  Write  five  consecutive  numbers  of  which  2  n  is  (1)  the 
largest,  (2)  the  smallest,  (3)  the  middle  number.  Add  the 
five  numbers  in  each  part.     Which  gives  the  simplest  result  ? 

38.  Write  five  consecutive  odd  numbers  of  which  2  n  +  1  is 
the  largest.  Write  five  consecutive  odd  numbers  of  which 
2  n  —  1  is  the  smallest. 

39.  Add  the  five  numbers  in  example  38,  first  part. 

40.  A  man's  salary  is  x  dollars.  How  much  was  it  5  years 
ago  if  it  has  been  increased  b  dollars  each  year  ? 

41.  A  machine  cuts  pieces  3  inches  long  from  a  rod  10  feet 
long.     How  much  is  left  after  x  cuts  ? 

42.  If  an  automobile  is  worth  m  dollars  and  depreciates  2  n 
dollars  in  value  the  first  year  and  n  dollars  each  succeeding 
year,  what  is  its  value  at  the  end  of  5  years  ? 

Evaluate  the  answer  if  m  =  1850,  n  —  231.50. 

43.  From  x4  +  3  ax3  —  2  bx2  -f-  3  ex  —  4  d  subtract 

3  x4  +  ax3  —  4  bx2  -f  6  ex  +  d. 
If  m=a2-f&2+c2,  w  =  a2+62-c2,.p=a2-&2+c2,  q=b2+c2-a2, 


alui 

ite  44  to  55  : 

44. 

m  4-p. 

48. 

m-\-?i+p+q. 

52. 

b2  +  p-q. 

45. 

0-p. 

49. 

a2-p  +  q. 

53. 

b2  -  n. 

46. 

m—n—p—q. 

50. 

m  —  p. 

54. 

m—n-\-p—q. 

47. 

m  —  n. 

51. 

m—n—p+q. 

55. 

p  +  q+P  +  q, 

PARENTHESES 

97.  In  algebra,  as  in  arithmetic,  it  is  frequently  necessary 
to  group  several  numbers  that  are  to  be  regarded  as  a  single 
number,  or  to  indicate  that  the  result  of  several  operations  is 
to  be  taken  as  a  whole.  The  parenthesis,  (  ),  is  generally  used 
to  inclose  such  a  group  of  numbers. 

Thus,  a2  +  b2  —  (2  a2  +  ab  -  5  b2)  means  that  the  expression  2  a2  +  ab 
—  5  b2  is  to  be  subtracted  from  a2  +  b2. 


58  Subtraction 

It  is  often  necessary  to  inclose  within  a  parenthesis  parts  of 
an  expression  already  inclosed  within  a  parenthesis.  For  this 
purpose  the  brackets,  [  ],  and  the  braces,  \  \ ,  are  used.  The  use 
of  the  vinculum,  ,  is  avoided  as  far  as  possible  on  account 
of  the  difficulty  in  printing  it.  All  these  symbols  have  the 
same  use  as  the  parenthesis  and  are  generally  referred  to  as 
parentheses. 

98.  It  has  already  been  explained  in  §  55,  that  in  a  series 
of  indicated  operations,  the  multiplications  and  divisions  are 
to  be  performed  in  the  order  given  before  the  additions  and 
subtractions. 

Thus,  12 +  3x5  =  12 +  15  =  27;   6-4-2x3  +  7x2  =  9+ 14  =  23. 

If  the  operations  are  not  to  be  performed  in  this  accepted 
order,  certain  numbers  of  a  series  may  be  inclosed  within  a 
parenthesis. 

The  operations  within  a  parenthesis  take  precedence  over 
all  others.  When  these  operations  are  performed  the  result- 
ing number  takes  the  place  of  the  parenthesis  in  the  series  of 
operations. 

Thus,  12  +  3  x  5  must  not  be  confused  with  (12  +  3)  x  5,  for  12  +  3 
x  5  =  27,  while  (12  +  3)  x  5  =  15  x  5  =  75. 

Also,  27-T-3  +  6  must  not  be  confused  with  27^(3  +  6),  for 
27  -s-  3  +  6  =  9  +  6  =  15,  while  27  -=-  (3  +  6)  =  27  -n  9  =  3. 

EXERCISE 

99.  Write  and  evaluate  examples  1  to  3. 

1.  76  diminished  by  the  sum  of  27  and  13. 

2.  25  increased  by  the  difference  between  23  and  6. 

3.  86  diminished  by  the  difference  between  118  and  97. 

4.  What  is  the  difference  in  meaning  between  the  expres- 
sions, a  —  b+c  and  a  —  (b  +•  c)  ? 

5.  What  is  the  difference  in  meaning  between  a—(b  —  c) 
and  a—b  —  c? 


Parentheses  59 

Evaluate  examples  6  to  10. 

6.  12 -7 -(2  +  1)  and  12-7-2  +  1. 

7.  12-(7-2  +  l)andl2-(7-2)+l. 

8.  12  _  7-(2  +  1)  and  12  -(7  -  2  +  1). 

9.  63  -(24  -  15  -  8)  and  63  -  24  -(15  -  8). 
10.    79 -(38 -17 -14 -2+ 9). 

100.  Removal  of  Parenthesis.  The  expression  5  +  (7  —  4) 
means  that  7  —  4  is  to  be  added  to  5  ; 

or5+(7-4)=5  +  3  =  8. 
.      But   5  +  7-4  =  8. 

...5  +(7_4)=5+7-4. 
Also  a+(2a  —  3&  +  c)    means   that   2  a  —  3  b  +  c   is  to  be 
added  to  a, 

or  a  +  (2  a  —  3  6  +  c)=  a  +  2  a  -  3  &  +  c 
=  3a-36  +  c. 

A  parenthesis  inclosing  any  number  of  terms  and  preceded  by  a  plus 
sign  may  be  removed  without  changing  the  signs  of  the  terms  inclosed 
in  the  parenthesis. 

ORAL  EXERCISE 

101.  Remove  the  parentheses  and  collect  the  terms  as  much 
as  possible  : 

1.  7 +(4 +  5).  6.    8a>+(10a-a;). 

2.  7d+(4<2  +  5d).  7.    18&+(7&-9&). 

3.  19/ +  (12/ +7?*).  8.    2r+(10-3r). 

4.  17x+(10x-y).  9.   (2x-3y)  +  (-x-y). 

5.  7*  +  (14* -5r).  10.    (5p-7g)+(-3p+5g). 

102.  The  expression  11  -  (8  +  2)  means  that  8  +  2  is  to  be 
subtracted  from  11, 

or  11  -(8  +  2)=  11  -  10  =  1. 
But  11  -  8  -  2  =  1. 
...  ii  _(8  +  2)=  11  -8  -2. 


60  Subtraction 

The  expression  3  a  -{-2  b  —  (a  —  3  6)  means  that  a  —  3  b  is  to 
be  subtracted  from  3  a  -+-  2  b.  The  rule  for  subtraction  is  : 
"  Change  the  signs  of  the  subtrahend  and  add  the  result  to  the 
minuend."  To  apply  this  rule  in  the  present  case  we  may 
remove  the  parenthesis,  changing  the  signs  of  the  terms  within 
the  parenthesis,  and  collect  terms. 

3  a  +  2  b  —  (a  -  3  6)  =  3  a  +  2 b  —  a  +  3  b 

=  2  a  +  5  b. 
Also,  3x2-x-(2x2-2x  +  7)  =  3x2-x-2x2  +  2x-7 

=  x*  +  x-7. 

The  student  may  verify  the  answer  by  ordinary  subtraction. 

3  x2  —  x 
2  x2  -  2  x  +  7 
x2  +  x  -  7 

A  parenthesis  inclosing  any  number  of  terms  and  preceded  by  a  minus 
sign  may  be  removed  provided  the  sign  of  each  term  inclosed  by  the 
parenthesis  is  changed. 

ORAL   EXERCISE 

103.  Remove  the  parentheses  and  simplify  as  much  as  possible : 

1.  32 -(17 +  6).  11.   17  r  -  (21  r-14r). 

2.  32 -(17 -.6).  12.  17r-(-21r  +  14r). 

3.  15  -(13  +  11).  13.  -(7  s  -13  s)  -15  s. 

4.  15  -  (13  -  11).  14.  -  (x  +  y)  — p  -J-  g. 

5.  a  -(b  -f-c).  15.  (a  — &)  +  (— a-f  6). 

6.  a—(b  —  c).  16.  (a-6)-(-a  +  6). 

7.  12p-(3i>  +  g).  17.  (2  +  3m)-(3-2ra). 

8.  12p-(-3p-q).  18.  (2-3m)  +  (3  +  2m). 

9.  15m  -(6m  +  2  m).  19.  23a -(16  + 5a). 
10.  15m -(-6m -2  m).  20.  23a +(- 16  -  5a). 
• 

104.  Sometimes  one  or  more  parentheses  are  inclosed  within 
a  parenthesis.     In  this  case  either  the  outer  or  the  inner  paren- 


Parentheses  61 

thesis  may  be  removed  first.     The  beginner  will  find  it  ad- 
visable to  remove  the  inner  parenthesis  first. 

=  18 x  —  {4y  —  [9aj  —  2 y  —  3*c  +  y]  j  (Removing  (  ).) 

=  l$x-\±y-§x  +  2  y  +  3x-y\  (Removing  [  ].) 

=  18  x  —  4  y  +  9  #  —  2y  —  3x  +  y  (Removing  \  \ .) 

=  24  x  —  5  y  (Collecting  terms.) 
EXERCISE 

105.   Simplify  by  removing  parentheses  and  combining   like 
terms. 

1.  (x-y-z)-(2x  +  y-3z). 

2.  25  -(3  +  4x2)  +  6. 

3.  1  +  ra  —  w—  (21  —  ra+2n). 

4.  (^-»)-(a;2-2a;  +  3)-(ar0  +  2a;-6). 

5.  (arJ  +  x)-(^-l). 

6.  x2-\-2ax  +  a2-(x2-2ax  +  a2). 

7.  8m— (4m  +  2  7i)  +  (5m  — 6n). 

8.  a2-(a2-2a6)  +  (-2a5  +  a2). 

9.  (4:p-q)-[2p-(q-p)+2p}. 

10.  y  +  [(m-»)  +  (m +/>)]. 

11.  y+[(m  +  n)-(w+j>)]. 

12.  y  —  [(m  — n)  — (j9  — n)]. 

13.  y  — [(ra  +  w)  — (ra— p)]. 

14.  7a-26-[(3a-c)-(26-3c)]. 

15.  2a-(3  6  +  2c) 

-{5&-3a-(a  +  &)+5c-[2a-(c-2&)]j. 

16.  16-a>-{-7»-[8-9a>-(3-6a)]|. 

17.  «4-;4^3-[6^-(4^-l)]j-(^  +  4^  +  6x2  +  4ir  +  l). 

18.  4.04 a  -  [.275  y-  (.5  b- 3.875  a) +3.6  y]  -(.165  a-. 375  y). 

19.  ab  -  [(3  6ce  -  2  a&)  -  (5  6ce  -  bef)  +  (3  06  -  3  6e/)]. 


62  Subtraction 

Simplify: 

20.  l-[_(2-^)]  +  [4^-(3-n^)]  +  4-(6x-5). 

21.  3m-38n-(o7p  +  15?)-(12|>-38?  +  48n-50m). 

1 06.  Inserting  parenthesis  : 

3  -  2  -f  3  =  3  +  (-  2  +3)  and  3  -  2  +  3  =  3  -(2  -  3). 
Also  a+b—c+d=a+b  +(  —  c  +  d) 
and  a  +  6  —  c-|-d  =  a-f-6—  (c  —  d). 
Let  the  student  verify  these  results  by  removing  the  paren- 
thesis according  to  the  rules  of  §§  100  and  102. 
From  these  results  we  draw  two  conclusions : 

1.  The  value  of  a  polynomial  is  not  changed  if  any  of  its  terms  are 
inclosed  within  a  parenthesis  preceded  by  a  plus  sign. 

2.  The  value  of  a  polynomial  is  not  changed  if  any  of  its  terms  are 
inclosed  in  a  parenthesis  preceded  by  a  minus  sign,  provided  the  sign  of 
each  term  inclosed  is  changed. 

EXERCISE 

107.  Inclose  in  a  parenthesis  preceded  by  the  plus  sign  the  3d 
and  4:th  terms  in  examples  1  to  5. 

1.  a  -f  b  —  c  +  m.  3.    am  -f  bx  —  ac  —  mx  +  2  b. 

2.  4  m  —  3  x  -f-  y  —  2  a  +  c.     4.   a2  +•  62  +  m2  —  2  mxc  +  2  a#. 

5.    ?/3  +  3  ax2  +  3  a2*  -fa3-  m2  +  &2. 

Inclose  in  a  parenthesis  preceded  by  the  minus  sign,  the  2d, 
3d,  and  4th  terms  of  examples  6  to  10. 

6.  p2  +  q  —  pq-  4-  m2  —  n2  —  3  mny. 

7.  m —  n—p -\-pq  —  l  y. 

8.  m2  +  2  raw  -f  w2-f-p2  —  2pq. 

9.  />3  —  3  p2g  -f  r/3  —  n3. 

10.  y2  +  2  po2  +  p2  -  n2  +  3  pg. 

11.  Inclose  in  a  parenthesis  preceded  by  the  minus  sign,  the 
last  three  terms  of  examples  6,  7,  8. 

12.  Inclose  the  last  four  terms  of  examples  8  and  10  in  a 
parenthesis  preceded  by  a  minus  sign. 


Equations  63 

EQUATIONS    INVOLVING    ADDITION,    SUBTRACTION,    AND 
PARENTHESES 

ORAL  EXERCISE 

108.  1.    If  4  x  4  3  x  =  7,  what  does  x  equal  ? 

2.  If  4  a;  —  3  x  =  7,  what  does  x  equal  ? 

3.  If  2  x  =  a;  4  2,  what  does  a?  equal  ? 

4.  If  3  x  4-  2  =  2  x  4-  3,  what  does  x  equal  ? 

/SoZtfe  Me  following  equations  and  check  the  results  : 

5.  a?4-2a;  =  3.  13.   8q  =  15+(—7). 

6.  2  a; -a;  =  5.  14.    16^4- 9  a;  =  50. 

7.  7j9-5p  =  6.  15.   #4-3  =  12. 

8.  7p  +  5p  =  12.  16.   2a;  +  6  =  12. 

9.  11  a; -7  a;  =  8 -4.  17.   2  v  +  5  v  =  21. 

10.  2  +  3a;  =  8.  18.   17»-7n  =  30. 

11.  2r  +  *5  =  6.  19.    15  a;  4  6  a;  =  42. 

12.  4m— 4  =  4.  20.   15  a  —  6a;  =  18. 

109.  Solve  5  a;  -(12  -a?)=  3  4(^4  3). 

Solution.    5x  -(12  —  x)  =  3  4(^4  3). 

5aj  —  12  4se  =  34a;4  3.     ( Removing  parentheses. ) 
6  x  —  12  =  6  +  x.  (Collecting  terms. ) 

5  x—  12  =  6.  (Subtracting    x     from     both 

members.) 
5  a;  =  18.  (Adding  12  to  both  members.) 

x  =  3f.  (Dividing  both  members  by  6.) 

The  result  may  be  checked  by  substituting  3f  for  x  and  simplifying. 
6x8|-(12-3|)  =  8+(8f  +  8). 
18  _  8f  =  3  +  6$ . 
9|  =  9f . 

EXERCISE 

110.  Solve  the  following  equations  and  check  the  results: 

1.  5a;- 1  =  14.  3.    7p-f  =  13J. 

2.  8a;4i  =  16i  4.   9^-5^  =  36. 


64  Subtraction 

Solve  and  check : 

5.  m-(-2m  +  3)=6.  10.  11  x  +  14  =  x-  16. 

6.  2s-5  =  4-(5-s).  11.  16c-(5  +  llc)=5. 

7.  5y-(3y  +  3)=3.  12.  5  a;  +  5  =  2x  +  6. 

8.  15a-10  a  =  45.  13.  11  n  —  23  =  7»  —  4 

9.  27&=31-(-20  6+4).  14.  J -(-J +  3)  =  3. 

15.  4y  —  11  =  3  —  y. 

16.  2a+(3  +  4)=a-(5  +  6). 

17.  m+(3+4m)=3m  +  3. 

18.  a;  — 14  =  14  — a:. 

19.  4cc  +  (2a  +  2)=2a;+(a}  +  l). 

PROBLEMS  SOLVED  BY  MEANS  OF  ALGEBRAIC 
EQUATIONS 

111.  If  there  are  two  unknown  numbers  to  be  found  in  a 
problem,  two  distinct  relations  of  the  numbers  must  either  be 
given  or  implied.  Generally  the  method  of  making  the  equa- 
tion is  as  follows : 

1.  Introduce  some  letter  as  x,  to  represent  one  of  the  unknown  num- 
bers, preferably  the  smaller  one. 

2.  Express  the  other  unknown  in  terms  of  x  by  using  one  of  the  two 
given  relations. 

3.  Make  an  equation  by  using  the  other  relation. 

EXERCISE 

112.  1.  The  sum  of  two  numbers  is  24,  and  one  of  them  is 
twice  as  large  as  the  other.     Find  the  two  numbers. 

Solution.     Let  x  =  the  smaller  number. 
Hence  2  x  =  the  larger  number. 
Then  x  +  2  x  =  24,     (By  the  first  condition  stated.) 
or  3  x  =  24. 
.  *.  x  —  8,  the  smaller  number, 
and  2  x  =  16,  the  larger  number. 


Problems  65 

2.  The  sum  of  two  numbers  is  18  and  one  is  five  times  as 
large  as  the  other.     Find  the  numbers. 

3.  The  sum  of  two  numbers  is  40,  and  the  larger  exceeds 
the  smaller  by  10.     Find  the  two  numbers. 

4.  The   difference   of   two  numbers  is  10,  and  the  larger 
number  is  3  times  the  smaller.     Find  the  numbers. 

5.  Find  two  parts  of  53,  one  of  which  exceeds  the  other  by 
11. 

6.  Find  two  parts  of  28,  one  of  which  exceeds  twice  the 
other  by  4. 

Suggestion.     If  x  =  the  smaller  part,  2  x  +  4  =  the  larger  part. 

7.  Find  two  numbers  whose  sum  is  23  and  whose  difference 
is  8. 

8.  Find  two  consecutive  numbers  whose  sum  is  73. 

Suggestion.     Since  the  numbers  are  consecutive  the  larger  exceeds 
the  smaller  by  1. 

9.  Find  two  consecutive  numbers  whose  sum  is  33. 

10.  Find  three  consecutive  numbers  whose  sum  is  33. 

11.  Separate  153  into  two  parts  of  which  the  larger  exceeds 
two  times  the  smaller  by  30. 

.12.   If  the  sum  of  two  consecutive  numbers  is  45,  find  the 
numbers. 

13.  A  rectangle  is  20  feet  longer  than  it  is  wide,  and  its 
perimeter  is  160  feet.     Find  its  length  and  width. 

Suggestion.  Let  x  =  the  number  of  feet  in  the  width. 

Hence  x  +  20  =  the  number  of  feet  in  the  length. 
Then  x  +  x  +  (x  +  20)  +  (x  -j-  20)  =  160. 
Let  the  student  solve  the  equation. 

14.  A  rectangle  is  twice  as  long  as  it  is  wide  and  its  perime- 
ter is  150  feet.     Find  its  length  and  width. 

15.  The  length  of  a  rectangular  lot  exceeds  twice  the  width 
by  50  feet  and  the  perimeter  is  364  feet.     Find  its  dimensions. 


66  Subtraction 

16.  If  n  is  the  middle  one  of  live  consecutive  numbers,  how 
would  you  represent  the  other  four  numbers  ?  Find  five  con- 
secutive numbers  whose  sum  is  45. 

17.  Three  men  divide  $300  so  that  the  second  has  $25  less 
than  the  first,  and  the  third  $50  more  than  the  second.  How 
many  dollars  does  each  man  get  ? 

Suggestion.     Let  x  =  the  number  of  dollars  the  first  receives. 

Hence  x  —  25  =  the  number  of  dollars  the  second  receives, 
and  x  +  25  =  the  number  of  dollars  the  third  receives. 
Then  x  +  (x  -  25)  +  (x  +  25)  =  300. 
Let  the  student  solve  the  equation. 

18.  The  combined  weight  of  the  largest  steam  locomotive 
and  the  largest  electric  locomotive  in  the  United  States  is  381 
tons.  The  steam  locomotive  weighs  57  tons  more  than  3  times 
the  weight  of  the  electric  locomotive.  What  is  the  weight  of 
each? 

19.  The  combined  cost  of  the  Panama  and  Suez  Canals  was 
approximately  394  million  dollars.  The  Panama  Canal  cost 
5  million  dollars  less  than  20  times  as  much  as  the  Suez  Canal. 
What  was  the  approximate  cost  of  each  ? 

20.  It  is  1274  miles  further  from  London  to  New  Orleans 
than  it  is  from  London  to  New  York,  and  the  sum  of  the  two 
distances  is  7740  miles.  Find  the  distance  from  London  to 
each  place. 

21.  Two  day-rate  telegrams  were  sent  from  New  York,  one 
to  Detroit  and  one  to  Winnipeg,  Manitoba.  The  two  messages 
cost  $  1.10.  The  message  to  Detroit  cost  35  cents  less  than 
the  one  to  Winnipeg.     Find  the  cost  of  each. 

Hint.     Change  $  1.10  to  110  cents. 

22.  Two  six-word  Marconigrams  (wireless  telegrams)  were 
sent  from  London,  one  to  New  York  and  one  to  St.  Louis. 
The  message  to  St.  Louis  cost  (in  United  States  money)  36 
cents  more  than  the  one  to  New  York,  and  the  total  cost  was 
$  2.10.     Find  the  cost  of  each. 


V.  MULTIPLICATION 

ORAL  EXERCISE 

113.  1.    What  is  the  law  of  signs  in  multiplication  ?    (§  49,  2.) 
Find  the  products : 

2.  (-8)(-2).2;  (-2)2;  (-2)'. 

3.  (-1)';   (-1)3;  (-1)^;   (-iy. 

4.  (-2)(-3)2;  (-2)*(-3);  (-2)2.3. 

5.  2  x  3  yd. ;  2  x  3  y ;  3  X  5  mi. ;  3  X  5  m. 

6.  4  x  10 a ;  5x8?/;  7  x  2 a  ;  4  x  3 ab. 

7.  a  •  ft  ;  a(-  6) ;  (-a)    b  ;  (-  a)(-  6). 

8.  7- (-3);  7- (-3  a);  5- (-4  a*/);  8-(-3a&). 

9.  _2.3;   -2-3a;  -4-76;  -5-2a6. 

10.  -4-  (-5);  -4- (-5a);  -4-(-7a6);   -J.(-5xy). 

11.  2.  (-3  6);  -4- 3  aft;   -  5  •  (-  9  x)  ;  4  .  (-  2  y). 

12.  3  x  4x(-2);  3x4x(-2a);   -4x36x2. 

13.  2x(-3)x4;  2x(-3)x(-4);   -  2  X  (-  3)  X  (-  4). 

114.  The  Law  of  Exponents  in  Multiplication.     Define  expo- 
nent and  base.     (§  64.) 

Since  22  =  2  .  2  and  23  =  2  .  2  .  2, 
therefore  22  x  2s  =  (2  •  2)  x  (2  .  2  •  2)  =  25  or  22+3. 
Similarly  a2  •  a3  =  (a  •  a)  x  (a  •  a  •  a)=  a5  or  a2+3. 
Similarly  a2  •  a2  •  a^  =  (a  -  a)  x  (a  -  a)  x  (a  -  a  -  a)  =  a7  or  a2+2+3. 
Also  since  am  =  a  •  a  •••  to  m  factors  and  an  =  a  •  a  —  to  ?i 
factors  therefore  aw  •  a"  =  am+n. 

67 


68  Multiplication 

The  equation,  am  •  a"  =  aw+",  is  the  law  of  exponents  for  mul- 
tiplication stated  in  algebraic  symbols.     In  words  we  have  : 

In  multiplying  powers  of  the  same  base  the  exponent  of  any  base  in 
the  product  is  equal  to  the  sum  of  its  exponents  in  the  factors. 

Also  since  (a3)2  =  a3  >  a?  =  a6,  we  have  (a3)2  =  a3X2,  and  in 
general  (am)n  =  amn. 

115.  The  student  must  note  that  the  law  am  •  an  =  am+n  ap- 
plies only  when  the  bases  are  the  same.  It  must  also  be 
remembered  that  when  no  exponent  is  written,  the  exponent 
1  is  understood. 

Thus,  x  •  x2  =  x1+2  =  x%. 

The  base  for  any  given   exponent  is    the  number   symbol 
immediately  preceding  it. 
Thus,  3  ab%  means  3  a  •  b  •  b. 

If  it  is  desired  that  the  exponent  shall  affect  other  pre- 
ceding numbers,  a  parenthesis  is  used. 

Thus,  (3a&)2  means  3  ab  x  3  aft,  and  is  read  "the  square  of  3a6." 
(a  +  by  is  read  "  the  square  of  the  binomial,  a  +  6." 

1.  x2-xz  =  xh.  4.    2-2"  =  2n+1. 

2.  (xy)*-(xyy=(xy)7.  5.    (x  +  2)2(a+2)5=(a;+ 2)7. 

3.  (_3)5.(_3)3  =  (_3)8#  6>      (a4)5  =  a20. 

ORAL   EXERCISE 

116.  Find  the  indicated  products : 


1. 

a2- 

X3. 

2. 

a3 

•  a4. 

3. 

ra2 

•  m5 

4. 

y7 

■f- 

5. 

p* 

■f. 

6. 

Qn 

•?<• 

7. 

r*. 

r13. 

8. 

b- 

b\ 

9. 

xp  -X. 

10. 

nr  •  n2. 

11. 

cn+1  •  c3. 

12. 

Xy  •  X2y. 

13. 

ax  •  ay. 

14. 

dm  •  dn. 

15. 

a2m  •  a2 

16. 

a2m  •  am 

a. 


17. 

m1x  .  mhx% 

18. 

X2  •  X  •  X3. 

19. 

m2  •  m3  •  m3. 

20. 

(a+6)2-(a+6)3. 

21. 

j*2  .  o*3  .  f^ 

22. 

(2  a)2  •  (2  a)3. 

23. 

(2  +  a)2.(2+z)6. 

24. 

43 .  42  •  4. 

Multiplication  of  Monomials  69 

MULTIPLICATION  OF  MONOMIALS 

117.  Remember  that  in  multiplying  two  or  more  monomials 
together : 

The  factors  of  a  product  may  be  arranged  in  any  order  without  chang- 
ing the  value  of  the  product. 

Thus,  2x3  =  3x2;  2x4x3  =  2x3x4;  etc. 

The  product  of  an  even  number  of  negative  factors  is  positive,  and 
the  product  of  an  odd  number  of  negative  factors  is  negative. 

Thus,  (—  a)  x  (—  b)  =  ab  ;  (—  m) (— m)(—  i»)  (—  m)  or  (—  m)4  = 
wt4;  (-<*)(-&)(- c)  =-abc;  (-a)3=-a3. 

1.   2a2b  x  (-3a364c)=2  -(-3)  .  a2 .  a3  •  b  •  b4  •  c  =  -6o555c. 
2.-5  ax2y  X  (-  3  x*yn)  =  (-  5)(-  3)a  •  a;2  •  ar»  •  y  •  yn 
=  15  aa^yn+1. 

118.  The  preceding  laws  together  with  the  law  of  exponents, 
§  114,  give  the  rule. 

To  multiply  monomials : 

1.  Find  the  product  of  the  numerical  coefficients,  keeping  in  mind  the 
law  of  signs  for  multiplication. 

2.  Write  after  this  product  the  product  of  the  literal  factors,  giving 
to  each  letter  an  exponent  equal  to  the  sum  of  its  exponents  in  the 
factors. 

Examples 

1.  (-3 a2b)  •  5 ab2c  =  - 15 a?bzc. 

2.  4  anb  •  anb2  =  4  a2n63. 

3.  anb .  ab  =  an+1b2. 

ORAL  EXERCISE 

119.  Find  the  products  of  the  following : 

1.  a2a2.  5.  4 #2.  3^.  9.  b2c2  •  b3c. 

2.  b2b4.  6.  5  a4- 3  a2.  10.  -62c2.(-63). 

3.  x*x\  7.  a2b>ab.  11.  3«&.ac. 

4.  c5c5.  8.  afy-ic3?/5.  12.  .v2a-xa. 


70  Multiplication 

Find  the  products  : 

13.  a2-a?>a\  22.  (- p)(- q)2r. 

14.  x  •  x2  •  x*.  23.  (—  xf  •  x. 

15.  2yy(-y2).  24.  (-M)4. 

16.  afr-afc2.  25.  (-  5)3. 

17.  3rs-3rs-rs.  26.  (—  4)4. 

18.  5pqx(-p2q2).  27.  -3a6(-c). 

19.  c2d(- cd2)(-c).  28.  -7j>(-  jd)(-  r). 

20.  x2f>2x*y\  29.  (2a)(-2  6)(-c). 

21.  (-p)(-q)(-r).  30.  a2(- a)3. 

EXERCISE 

120.   Find  the  products  of  the  following : 

1.  x2tf-2x*y\  9.    (-2 ma) (1.26V). 

2.  ia4(-|a;7).  10.    .32#2/(-llary). 

3.  3a- 2a2- 6a3.  11.   (-  80maf)  (.05m?x). 

4.  a26(-3a3c)(-63c2).  12.    (-  .2x*y2)  .  hxy\ 

5.  3(a+&)2-(a  +  &)3-(-4>      13.    2&a(- 5.5&Y). 

6.  '5oP#-2n&.  14.   12  a2z  •  J  aa4. 

7.  ( _  31  aty)  .  e  a2/5#  15.   .33^  m2n  .  15  mws .  m7l. 

8.  (-3a2)x(-7aa5).  16.    £px*  •  5byx2. 

17.  (-15a2rc).3&.(-22a6)2a2&. 

18.  (-3ac)(.33|ay)(acv). 

19.  .4  6..2  62c-26c2.  25.  (- 40awl)(- .05  a*). 

20.  4ia2c3.2|ac4.1Va4c.  26.  m""2  •  m2. 

21.  am'an-a?.  27.  .82r"4(-  .4y). 

22.  ax+1az+2.  28.  c*  •  d'"1  •  c  •  d2"v. 

23.  yhf.  29.  cTdy+1  •  cd2+I/. 

24.  3an-|a.  30.  xn~2ym-\- xn^ym'1). 


Multiplication  of  a  Polynomial  by  a  Monomial     71 

31.  (  -  \  «m41)  •  7  a  •  2  a1""*.         39.  (f^f*4  •  3  c2-^4-*. 

32.  i5d'"-n  •  2d2"-m.  40.  —  mx_2w3  •  lOm3"^^. 

33.  (-  bpc<)  •  9  6«cp.  41.  3(x2y(-2xy2). 

34.  am+"  •  am-».  42.  3  (2  a;2)2 .  (-  3  a;). 

35.  5  a1"2*.  3  a4*.  43.  4(2af)3(-  4  a2). 

36.  y2zp-3y2n-2-zn-p.  44.  6  (3  #2)2(  -  4  a2)2. 

37.  (-2aw+2n).f  a4"w.  45.  6(2a&)2(-  a2b)2. 

38.  a4na2w  •  a?x2  •  anam.  46.  5(3  xy)\  —  xy2)2. 

MULTIPLICATION  OF  A  POLYNOMIAL  BY  A   MONOMIAL 

121.  4(3 +  5)=  4- 8  =  32. 

This  result  might  have  been  found  by  multiplying  the  num- 
bers within  the  parenthesis  separately  by  4. 
Thus,  4(3  +  6)=  4  •  3  +  4  .  5  =  12  +  20  =  32. 

53 

j  Ordinary  arithmetical  multiplication,  if  done  without  the 

m       abbreviating  process  of  "carrying,"  shows  the  same  principle. 
The  multiplication  at  the  left,  if  written  in  a  line,  is 
§52  7(50 +  3)  =  350 +  21  =  371. 

371 

The  algebraic  law  that  covers  this  case  may  be  expressed 
in  algebraic  symbols  thus, 

a  (b  -f-  c)  =  ab  +  ac. 

122.  In  words  this  law  gives  us  the  following  rule : 

To  multiply  a  polynomial  by  a  monomial,  multiply  each  term  of  the 
polynomial  by  the  monomial  and  unite  the  results  with  their  respective 
signs. 

Examples 

1.    Multiply  2  a2b  -  b2c  +  c2  by  -  3  a2b2. 

Multiplication         Check,    a  =  b  =  c  =  2. 
2  a2b  -  b2c  4-  c2  =12 

-  3  a*b2 ==-48 

-  6  a463  +  3  a264c  -  3  a*62c2  =  -  576 


72  Multiplication 

2.   Multiply  am~l  -f  bn~l  —  cp~l  by  —  abc. 
am~i  +  bn~l  —  c*-1 

—  abc 

—  ambc  —  abnc  +  abcp. 

ORAL  EXERCISE 

123.  Multiply  the  following  : 

1.  a(6  +  c).  11.  —  5x(x  +  y  +  z). 

2.  —  x(a  +  6).  12.  —  5  x(—  x  +  y  —  z). 

3.  —  a(x  —  ?/).  13.  —p(pq  —  r). 

4.  2  6(3  a -f  c).  14.  r(st-rs). 

5.  4  a2(3  x2  -  y).  15.  pq(p2q  -  pq2). 
6.-3  a2(2  a  +  6).  16.  a\bc  —  ac  +  a). 

7.  £c(a  +  &  _|_  c).  17.  -3h{k-hk  +  h2k2). 

8.  —  #(a  —  6  —  c).  18.  2  x(xy  —  xz  —  yz). 

9.  —  3x(a  +  b—  c).  19.  xy(xn~i  —  yn~l). 

10.    -4c(-2c  +  3d-6e).  20.  -  3x2y(xn~2y  -  xyn~l). 

EXERCISE 

124.  Multiply  the  following  : 

1.  r*(r  +  s).  4.    -3a6(a2  +  62). 

2.  a2(a&  —  ac).  5.    —  4  x(2  x  — .  5  y  —  3  z). 

3.  -2aj(a2-2a-l).  6.    -3«262(a  +  6  +  c). 

7.  a(«2  +  &2 +  <*&)• 

8.  .2  m(.2m2  +  .02  mn  +  .002  n2)n. 

9.  -3a(-4a2-f-2rc -i). 

10.  -  3  x2y2(2  xhf  -  3  x2y2  +  4  a^3). 

11.  3  ra2(2  ra3  -  7  ??i2  —  m). 

12.  -x-2/(21aj22/2-14^/  +  7)(-l). 

13.  3  par(  —  pq  —  5}or  —  7  or). 

14.  52(5  +  52  +  53). 


Multiplication  of  a  Polynomial  by  a  Monomial     73 


15.  _9a6c(-|a^|6-|c} 

16.  5-137  =  5(100  +  30  +  7)=? 

"•  S-f»-«X-W 


Multiply  the  folloioing : 

18.       3a2-62  +  7c2 


-  2  a%W 

19. 

6^-3^-9^  +  18 
.3x 

20. 

.6x3-.$x2y  +  2xy*-2y3 
.bxY 

21. 

2a26-3cd3  +  £ac3 
-6acW 

Simplify  the  following  : 

22.  4:(2x-7y)+2(x  +  14:y). 

23.  4a(a&  +  6c  +  ca)-2  6(a2  +  2ac). 

24.  «(«2-a^  +  2/2)+2/(^2-^  +  2/2)- 

25.  a(a2  +  «2/  +  2/2)-  y(x*  +  xy  +  ?/2). 

26.  (or*  -  3  afy)*/3  ~(yz  -Sxy2)  x*. 

27.  12(i^-i2/  +  i2)-16(^  +  i2/-i2). 

28.  a-2[3a-6-2(6-a)  +  3(a-26)]. 

29.  2x-8z-3[2y-(2x-z)]-3(x-y-z). 

30.  4a(6-3)-56(a-2)+a6  +  7(a-6). 

31.  a(b  —  a  +  d)  —  6(a  +  c  -  d)+c(a+  b  —  d). 

32.  3 -6a(6-c)-26(9a-c)-2c(6-9c). 

33.  p[q(s  +  *)  -  tf]  -  «(pg  -  1)  +  *(4  -  pq)  +  j9S*. 

34.  (8c2  +  24cd3-12c2.r-3)|cnd». 

35.  (3&2-6c2  +  9&c)(-p*c*). 


74  Multiplication 

Simplify  : 

36.  (7  an  -  3  a*"1  -  2  a*"2) ( -  .4  a""2). 

37.  (9  xp?/«  —  4  x^y*'1  +■  3  x^yi-^y2. 

38.  (8  a1"2-  +  &3_n)(-  5  a3m6"). 

39.  (#m  +  yp  +■  z«)ajy». 

MULTIPLICATION  OF  A  POLYNOMIAL  BY  A  POLYNOMIAL 

125.    1.    To  multiply  32  by  4,  we  may  first  multiply  2  by  4, 
then  multiply  30  by  4,  and  add  the  partial  products. 

Thus,  32  =    30  +  2 

4  =  4 

128  =  120  +  8. 

2.    To  multiply  32  by  24,  we  may  first  multiply  30  +  2  by  4 
and  then  by  20,  and  add  the  partial  products. 

Thus,  32  =  30  +  2 

24=  20  +  4 

128  =  120  +  8 

64    =  600  +    40 


768  =  600  +  160  +  8. 


3.   To  multiply  2a  +  36  by  3  a  4-  b,  we  first  multiply  2  a  + 
3  b  by  3  a  and  then  by  6  and  add  the  partial  products. 

Thus,  2  a  +3  b 
3a  +    b 


6  a2  +    9  ab 
2  aft  +  3  62 

6  a2  +  1 1  a&  +  3  62. 

4.    Multiply  or5  -  2z2  -  3x  +  2  by  2  a?  -  3. 

An    orderly    arrangement    of    the 
X3  —  2  X-  —  3x-\-    2  terms  of  the  polynomial  and  of  the 

9      o  partial  products  is  desirable.      It  is 

— — a    2   i — A~  usual  to  arrange  in  descending  powers 

x  ~         ~~  """  of  some  letter  (x  in  the  present  case), 

—  3ar  +-  off  +-    \)x  —  b       an(j  t0  arrange  the  terms  of  the  partial 

2a?4  —  7a?  +13 a;  —  6.      products  with  like  terms  in  the  same 

vertical  column. 


Multiplication  of  a  Polynomial  by  a  Polynomial    75 

126.  These  examples  lead  to  the  rule. 
To  multiply  one  polynomial  by  another: 

1.  Arrange  the  terms  in  descending  powers  of  some  letter  (or  alpha- 
betically). 

2.  Multiply  the  multiplicand  by  each  term  of  the  multiplier,  writing 
like  terms  in  the  same  column. 

3.  Add  the  columns  of  like  terms  and  join  the  results  obtained  with 
their  respective  signs. 

127.  To  check  the  answers  we  may  proceed  as  in  addition 
(§  84),  by  using  arbitrary  values  of  the  letters. 

Examples 

1.  Multiply  x2-3x-2hj  x  +  5. 
Multiplication  Check,     x  =  2. 

a?-3x  -2  =-    4 

x  +  5 = 7 

x^^Sx2-    2x  -28. 

5a2  -15a;  -10 
ic3  +  2a;2-17a;-10  =  -28. 

Substituting  1  for  x  in  the  above  example  will  not,  check  the  exponents. 
(Why  ?)     Hence  it  is  better  to  use  some  other  small  number,  as  2. 

2.  Multiply  3  a36  -  2  a2b2  +  ab3  by  2  a2  -  5  b2  -  ab. 
Multiplication  Check,    a  =  b  =  2. 

3  a36  -  2  a2b2  +  ab3  =       32 

2a2    -ab       -5b2  =-16 

6ft56-4ft462+    2ft363  -512. 

-3ci462+    2a363-      a?b4 

-15ft363+10a2//-5ftfr5 

6  abb  -  7  aAb2  -  11  aW  +  9  a2b4-5  a&5  =  -  512. 

EXERCISE 

128    Arrange  conveniently  for  multiplication  : 

1.  (3x*  —  ±y2x  —  x2y  +  4  f) (2  xy  -  3  y2  +  x2). 

2.  (4ft62-h6  63-3a26-h7ft3)(62  +  a6  +  ft2). 

3.  (3-ar»  +  2a;2-x)(a;-3x2  +  2). 


76  Multiplication 

Multiply  the  followiny  : 

4.   a  +  3  6.    c  -  2  8.    r  +  2  10.    2  a;  +  1 

a  — 5  c  —  6  r  +  s  2x  —  1 

ic  +  2/  9.    m  +  n  11.    a;  +  1 

x  —  y  m  +p  x  +  1 

15.  2  62  -  2.4  6  +  1.6 

106  +20 

16.  a  +  6  —  2 
q-  6  +  2 

17.  m2  —  mn  +  n2 

m  +  n 


5. 
12. 

a; +  3            7. 
x  +  2 

a? -a +  2 
a  +2 

13. 

x2  +  a;  +  1 
a>  -1 

14. 

2  a2  -3a  -5 

5a  -7 

18.  (2x2  +  l)(2a;2-l).  22.    (2  a;  +  3)(4a;2-  6a;  +  9). 

19.  (x  +  1)2.  23.   (3aj-7)(3a?  +  7). 

20.  (a;-4)2.  24.    (2  R  +  3)  (72  -  1). 

21.  (»  +  l)(a;  +  3).  25.    (2  m  +  £>)(-  2  m  +p), 

26.  (l-3x2  +  x)(x2  +  l-x). 

27.  (5a2-3a6-2  62)(a2  +  2a6). 

28.  (x2  +  xy +  y2)(x2-xy  +  y2). 

29.  (3a1  -  5  a6  -  2  62)(a2  -  7  ad). 

30.  (a;2  +  7a;-5)(aj2  +  5-7a;). 

31.  (af-5aa;-2a2)(a;2+ 3aa;  +  2a2). 

32.  (c4  -  c2)(c*  +  c). 

33.  (7p2-3g2)(4p2  +  ?2). 

34.  (x4  —  afy  +  x2y2  —  xy3  +  2/4)(a?  +  y). 

35.  (m2  +  n2  + 1)2  —  mw  —  m/>  —  np)(m  +  w  + 1>). 

36.  (3  a;2  —  2  a^  +  ?/2)(3  a;2  +  2  an/  -  ?/2). 

37.  (a3-aV  +  a^-x9)(o  +  4 

38.  (8a3  +  4a26  +  2a62  +  63)(2a-6). 

39.  (a  +  6  +  a;  +  y)(a  +  6  —  x  —  ?/). 

40.  (a  +  6  +  c  +  fi)(a-6  +  c-d). 


Multiplication  of  a  Polynomial  by  a  Polynomial    77 

41.  (a?  +  l)(.r2  +  2)(:c2  +  3). 

42.  (2a?-3)(3«  +  7)(6«-  5), 

43.  (3z  +  5)(7a-  +  5)(2a;-l). 

44.  (3  a  +  2  6)(a  -6)+(4a  +  5  6)(2  a  +  3  6). 

45.  (w-fv)(2i;-w)-f(M-t;)(H2M). 

46.  (x  +  ±)(x-2)-(x  +  2)(x-l). 

47.  (3  a  +  5)(2  a  -  3)(a  -  l)-(x  -  l)(a  +  2)(ic  -  3). 

48.  (a  +  b)  (c  -  d)  -  (a  -  6)(c  +  d). 

49.  (2  aa2  +  3  a#2)(2  az2  -  3  ay2). 

50.  (4  b2x2  +  5  ch/2)2.  55.    (aj°  +  #*)2. 

51.  («  +  6)3.  56.    (af  +  a;)2. 

52.  (Sx  +  Ayf.  57.    (a;2a  +  af  +  l)(af  -  1). 

53.  (3  aa  -  4  6?/)3.  58.    (aB+1  +  aB  +  a"-1)(a2  +  a). 

54.  (a*  -  6w)(a*  +  &").  59.    (a2n  +  2  an  +  4)  (an  -  2). 
In  examples  60  to  71, 

A  =  x*-2x  +  4,B  =  x>  +  2x  +  4,C=x-2,D  =  x  +  2. 

Perform  the  indicated  operations: 

60.  A-B.         64.    £  •  C2.  68.  Z)2  -  A 

61.  A  •  Z>.  65.    B2.  69.  C2  -  £. 

62.  A2.  66.    C2  •  Z)2.  70.  fe  -  C3. 

63.  A -IP.        67.    A'D-BC.  71.  Zte  +  Ca;  -(-4  +  B). 

72.  From  the  product  of  x  —  3  times  a;2  -f  a;  —  2,  subtract  the 
sum  of  x3  -f-  5,  and  3  x2  —  7  a?. 

73.  Multiply  2  a;  -  a,-3  +  7  by  the  sum  of  7  —  a;  and  2  a;  —  10. 

74.  Solve  7(x  +  2)  -  3(x  -  1)  =  2(a>  -  1)  +  25. 
Solution. 

7(se  +  2)-  3(x  -  1)  =  2(x  -  1)  +  25. 
7z  +  14-3z  +  3  =  2z-2  +  25.        (Why?) 

4  x  +  17  =  2  x  +  23.  (Collecting  terms. ) 

2  z  +  17  =  23.  (Subtracting  2  z   from  both 

members  of  equation.) 
2  a;  =  6.  (Subtracting    17    from   both 

members  of  equation.) 
x  =  3.  (Why?) 


78  Multiplication 

Solve  the  following  equations  : 

75.  2(a  +  5)  =  20. 

76.  2(z +  2)+ 3(3  a -3)  =  6. 

77.  S(x-  l)+7  =  11. 

78.  2(r  -  1)  +  2(r  -  2)  =  3(r  +  3). 

79.  3(^-5)4-8  =  18. 

80.  15(x  -  3)  -  17  =  103. 

81.  8(5  x-  37)-  4(3  x  -17)=  20. 

82.  6(a?-5)H-2a?  =  6aj-2(aj-f  10). 

83.  17(m-17)-17  =  -51. 

84.  18a-2(3  +  5a)=10. 

85.  7(3 p  -  2)  +  5(p  -  3)  -  4(p  -  17)=  110. 

86.  3(aj-5)-  4(a>-2)  +  6  x  =15. 

TYPE  FORMS  IN  MULTIPLICATION 

129.  Certain  multiplications  occur  so  frequently  that  it  is 
helpful  to  be  able  to  write  the  products  at  sight.  Seven  such 
special  products  are  given.  They  form  a  sort  of  algebraic 
multiplication  table,  and  should  be  thoroughly  learned  and 
understood. 

The  Square  of  the  Sum  or  of  the  Differ- 

II.   a  -  b 
a  —  b 


130. 

Types  I  and  II. 

ence  of  Two  Numbers. 

I.   a  +b 

a  +& 

a2  H-     ab 

4-     ab  4-  b2 

a2  —     ab 
—     ab-\-b2 

a2  4-  2  ab  4-  b2  a2-2ab  +  b2 

(a  4-  b)2  =  a2  +  2  ab  4-  ft2.  (a  -  6)2  =  a2  -2  a&  4-  V. 

Type  I  may  be  stated  in  words  : 

The  square  of  the  sum  of  two  numbers  equals  the  square  of  the  first 
plus  twice  the  product  of  the  first  by  the  second  plus  the  square  of  the 
second. 


ab 

62 

a* 

ab 

Type  Forms  in  Multiplication  79 

Let  the  student  state  Type  II  in 
words. 

The  product  (a  +  b)2  =  a2  +  2  ab  +  b2 
may  be  represented  by  a  figure.  a 

Examples 

1.  (2  x2  +  3)2=(2  x2)2+  2(2  z2)  •  3  +  32 

=  4  x*  + 12  x2  +  9.  a  6 

In  applying  the  type  form  to  find  the  square  of  the  binomial 
2  x2  +  3  we  note  that  the  a  of  the  type  is  to  be  replaced  by 
2  x2,  and  b  by  3.  Therefore  in  the  second  member  of  the  equa- 
tion, (a  +  b)2  =  a2  +  2  ab  +  62,  we  shall  put  4  a4  for  a2,  12  x2  for 
2  a6,  and  9  for  b2. 

2.  (2  a  -  3  ?/)2  =(2  x)2  -  2(2  s)(3  y)  +  (3  y)2  =  ±x2-12  xy 
+  9  y\     Explain. 

3.  (x2  —  ±y)2  =  xA  —  8x2y  +  16y2.     Explain. 

4.  132  =  (10  +  3)  2  =  100  +  60  +  9  =  169.     Explain 

5.  (-a  +  6)2  =  [(-a)+  6]2=  a2  -  2  a& +62.     Explain. 

6.  (-a-6)2  =  [(-a)+(-6)]2  =  a2  +  2a&  +  &2.     Explain. 

EXERCISE 

131.  Square  the  following  binomials  by  inspection,  using  Types 
I  and  II : 

1.  (x  +  y)2;  (x-y)2;  (y  -  x)2. 

2.  (c  -  a)2 ;  (c  +  a)2 ;  (a  -  c)2. 

3.  (r  +  s)2;  (r-s)2;  (-r  +  s)2. 

4.  (m  +  n)2;  (m  —  rc)2;  (—  m  —  n)\ 

5.  O  +  9)2;  0>-?)2;  fe-p)2- 

6.  522  =(50  +  2)2 ;  482  =  (50  -  2)2. 

7.  252  =  (20  +  5)2 ;  252  =  (30  -  5)2. 

8.  (a  +  2)2;  (a-2)2;  (2  -  a)2. 

9.  (4+6)2;  (4-6)2;   (6-4)2. 


80  Multiplication 

10.  (z  +  5)2;  (a -5)2;  (_x-5)2. 

11.  (6  +  n)2 ;  (6  -  n)2 ;  (-  6  -  n)\ 

12.  (a+7)2;  (a; -7)2;  (7  -  a;)2. 

13.  (a:2 +  3)2;  (aj2-3)2;  (3-af)* 


14. 

(2 

a+36)2;  (3a-26)2;  (26-3a)2. 

15. 

(az  +  y)2;  (ax  +  y2)2 ;  (a»  +  2Z3)2. 

16. 

(3  a  +  2)2. 

22.    492  =  (50  - 1)2.    28.    (5  a3  -  2  64)2. 

17. 

(4  +  2/)2. 

23.    (-a  -by.            29.    (_9p-3g)2. 

18. 

(8  -  m)2. 

24.    (4a-56)2.           30.    (2^  +  3)2. 

19. 

(a*/  +  z)2. 

25.    (a -lO)2.              31.    (m2-^)2' 

20. 

(_2  +  x2)2. 

26.    (7c-4d2)2.         32.   (7  — 4a£)». 

21. 

972  =  (100-, 

3)2. 

27.    (2  w  +  u)2.             33.    (7  ahj  -  3  aa)2. 

34. 

(20i)2=  (20+^)2 

37.   (2z  +  $yy. 

35. 

(3  6c  -  2  cd)2 

38.    322  =  (30  +  2)2. 

36. 

(1«-^)2. 

39.    652  =(60 +5)2. 

Of  what  binomial  is  each  of  the  following  trinomials  the  square  ? 

40.  x2  +  2xy  +  2/2.  45.  9  m2  -  24  mp  +  16p2. 

41.  m2  +  12rap  +  36p2.  46.  9  -  6  a  +  a2. 

42.  a;2 +  4 a +4.  47.  4a2  — 4a +  1. 

43.  r2  -  14 r  +  49.  48.  a;4  +  2a;2  +  1. 

44.  x2y2  —  6a^  +  9.  49.  x2y2  — 16  xy  +  64. 

132.    Type  III.     The  Product  of  the  Sum  and  the  Difference  of 
Two  Numbers. 

a  +  6 
a  —  b 
a?  +  ab 

-ab-b* 
a2  -62 

(a  +  6)(a-6)=a2  -  G2 


Type  Forms  in  Multiplication  81 

Type  III  may  be  stated  in  words  : 

The  product  of  the  sum  and  the  difference  of  two  numbers  equals 
the  square  of  the  first  minus  the  square  of  the  second. 

Examples 

2.  (3*  -  5y)  (Sx  +  5y)  =  (3x)2  -(By)*  =  9x2  -25  y\ 

In  this  example  a  of  the  type  form  is  replaced  by  3  x,  and 
b,  by  5  y.  Therefore,  in  the  product  we  shall  have  to  replace 
a?  by  (3  x)2,  and  b2  by  (5  y)2.  This  will  give  the  result  obtained 
above. 

3.  (x  +  2  y)(2  y  -  x)  =  (2  y  +  x)(2y  -  x)=±y2  -  x2. 

4.  (-  2  +  *)(-  2  -  *)=[(-  2)+  «][(■-  2)-  a]=  4  -  aft 

EXERCISE 
133.    Multiply  by  Type  III: 

1.  (x  +  y)(x-y).  11.  0  +  2?)(p_2?). 

2.  (c  +  d)(c  — d).  12.  (7i-f  5fc)(ofc-ft). 

3.  (r  +  s)(r-s).  13.  (l  +  4m)(4m-l). 

4.  (a  +  5)(a-5).  14.  (2a+36)(2a-36). 

5.  (4  +  «)(4-a).  15.  (3*-fr2y)(2y-3*). 

6.  (2a  +  6)(2o-6).  16.  22  xl8  =  (20+2)(20-2). 

7.  (a*  +  2)(tf-2).  17.  27x33=(30-3)(30+3). 

8.  (3«2  +  2)(3a2-2).  18.  49x51. 

9.  (2x  +  y)(2x-y).  19.  68x72. 
10.    (3a  +  c)(3a-c).  20.  103x97. 

In  ivhich  of  the  examples  21  to  28,  may  we  apjily  Type  III  ? 
Give  reason  in  each  case. 

21.  (2a  +  3  6)(2a-36).  25.    (-a +  &)(&  + a). 

22.  (2  a  +  6)(2  a2  -  6).  26.    29  x  31. 

23.  (x  +  y)(x2  -  y2).  27.    25  x  35. 

24.  (1  +  x)(\  —  »).  28.    (m  -  ?i)(a  -  6). 


82  Multiplication 

Multiply : 

29.  (a2  — 6)(a2  +  b).  33.  (ab  +  cd)(cd  -  ab). 

30.  (2*-z2)(2;c+z2).  34.  (3  a2  -  6)  (3  a2  +  6). 

31.  (im  +  ir)(im-ir).  35.  (-  2  +  3a)(2  +  3 a). 

32.  (2a3-62)(2a3  +  62).  36.  87-93. 

37.    (a  +  6  +  c)(a  +  6  —  c). 

Solution,     (a  -j-  6  +  c)  (a  +  6  —  c) 

=  [(o  +  6)+c][(a  +  6)  -c] 

=  (a  +  b)2-c2 

=  a2  +  2  a&  +  &2  -  c2. 

38.    (a-  b  +  c)(a  +  b  +  c). 

Hint,     (a  -  b  +  c)(a  +  b  +  c)  =  [(a  +  c)—  6][(a  +  c)  +  6],  etc. 

39.    (a  —  &  +  c)(a  +  6  —  c)  =  [a  —  (&  —  c)][a  +  (6  —  c)],  etc. 

40.  (a  —  6  +  c)(a  —  b  —  c). 

41.  (m  —  n  +p)(m  ■+-  w  —  p). 

42.  (/)-29  +  3r)(j9-f-2g-|-3r). 

43.  (2a;-y-3«)(2a;-y  +  3z). 

44.  (2x  —  y  —  3z)(2x  +  y  +  3z). 

45.  (x2  +  y2  -\-xy)(x2  +  ?/  —  xy). 

46.  (a-^)(a  +  ^)(a2  +  ^2). 

47.  (x  -  y)(x  +  y)(x2  +  y2)(xA  +  y4). 

48.  (^+2/n)(«n-2/n)- 

49.  (m*  —  rn^Xm2*  +  m*>). 

50.  (a?'  +  p2)(a;  +  p)(a;  — #). 

51.  [(a-l)(a+l)]> 

52.  (2  6-c)(2  6  +  c)+(c-2  6)(c  +  2  6). 

53.  [(x  +  y)(x-y)Y  +  [(y-x)(y  +  x)J. 

54.  (2r-3o2)(3a2  +  2r). 

55.  (-r-Ss)(r-?>s). 

56.  (2r2-4s)(-4s-2^). 


Type  Forms  in  Multiplication 


83 


Write  each  of  the  following  expressions  as  the  product  of  two 
binomials  : 

57.  a2  — 4.  61.    x2y2-a2b2. 

58.  9a2-16  62.  62.    xhfz2  -  25. 

59.  36 -Six4.  63.    25-16. 


60.   4a2-16  62. 


64.    x2n 


134.     Type  IV.     The  Product  of  Two  Binomials  having  a  Com- 
mon Term. 

The  two  binomials  are  of  the  form  x  +  a  and  x  +  b.     By 
multiplying,  and  arranging  in  order  of  powers  of  x,  we  have 
x  +  2  x  +  a 

x  +  3  x  +  b 


x2  +  2x 
3a  +  6 


x2  +  ax 

bx  +  ab 


x2  +  5x  +  6.  x2  -f-(a  +  b)x  +  ab. 

This  gives  (x  +  a)(x  +  b)  =  x2  -f  (a  +  &)*  +  a&. 

In  words :  x 

The  product  of  two  binomials  having  a  com- 
mon term  equals  the  square  of  the  common 
term,  plus  the  product  of  the  common  term  by 
the  sum  of  the  other  terms,  plus  the  product 
of  the  other  terms. 

The  product  (x  +  a){x  +  b)  =  x2  +  (a  +  b)x 
+  ab,  which  is  the  same  as  x2  +  ax  +  bx  +  ab, 
may  be  represented  by  the  figure. 

Examples 

1.  (x  +  3)(x  +  7)  =  x2+10x  +  21. 
Here  a  =  3  and  b  =  7,   a  +  6  =  10  and  a&  =  21. 

2.  (a?  -  3)(ar+  2)  =  a;2  +  [(-  3)  +  2>  +(-3)  X 2=x2-z-6. 

3.  (x  +  &)(aj  -  2)  =  .T2  +  (6  -  2)x  -  2  6. 

4.  (.T  —  p)(o;  —  r)  =  x2  —  (/)  +  r)<B  -f  /)r. 

5.  (m2-3p)(m2  +  5p')  =  mi  +  2m2p-15p2. 


bx 

ab 

X* 

ax 

84 


Multiplication 


EXERCISE 


135.    Which  of  the  following  can  be  multiplied  by  Type  IV 
Give  reason  for  each  answer. 


1. 

(a  +  7)(a  +  3). 

6. 

(5  +  a)(5  +  b). 

2. 

(a2-7)(a  +  3). 

7. 

(5  + «X6  +  6). 

3. 

(m  +  2  p)(m  -f  2  p). 

8. 

(5  +  a)(6  +  a). 

4. 

(3a+7)(4a+10). 

9. 

(5  +  a)(5  -  a). 

5. 

(3  a  -  7)(3  a  +  7). 

10. 

(3a2  +  5)(3a2-  7). 

Multiply  by  Type  IV: 

11. 

(*  +  7)(»+3). 

31. 

(a  +  6)(a  +  2  6). 

12. 

(a-2)(a-3). 

32. 

(a  -  6) (a  +  2  6). 

13. 

(x-\)(x-§). 

33. 

(a  +  6)(a-2  6). 

14. 

(x  +  l)(x-G). 

34. 

(a  —  2  x)(a  —3x). 

15. 

(x-  l)(ar +  6). 

35. 

(a  +  2x)(a  —  3x). 

16. 

(x+3)(x-2). 

36. 

(r  +  2s)(r  +  5s). 

17. 

0_3)(*  +  2). 

37. 

(r-  2s)(r  +  5s). 

18. 

(a?  +  3)(*  4-  2). 

38. 

(p  —  2rs){p  —  3rs). 

19. 

(a?  +  4)(a?  -  5). 

39. 

(ra  +  5r2)(m  +  2r2). 

20. 

(a  +  8)(a-5). 

40. 

(m-5r2)(m  +  2r2) 

21. 

(a  +  3)(a  +  2). 

41. 

(62_7)(62+12). 

22. 

(a  +  3)(a-2). 

42. 

(a2-3)(a2-6). 

23. 

(a  +  6)(a+l). 

43. 

(a  +  c)(a  —  d). 

24. 

(«  +  6)(a-l). 

44. 

(2a  +  3)(2a  +  5). 

25. 

(r- 3)(r-5). 

45. 

(3a2-5)(3a2-12). 

26. 

(r-8)(r  +  5). 

46. 

(3a?-5jf)(3&+5t/r). 

27. 

(r  +  3)(r-5). 

47. 

(3x2-5y2)(3x2+12  2/2). 

28. 

(a2-9)(a2  +  5). 

48. 

(a?/-5)(^  +  7). 

29. 

(a2  +  9)(a«  -  5> 

49. 

(ay  — 55)(o#  +  l). 

30. 

(62-  3)(&2-2). 

50. 

(xy  —  5  z)(#y  —3  2). 

Type  Forms  in  Multiplication  85 

51.  (2  x  +  7  y)  (2  x  +  3  y).  56.  (10  +  2)(10  +  8). 

52.  (2x  +  7y)(2x  +  5).  57.  (60  +  2)(60  +  1). 

53.  (ar  +  5  x)(ar  -  7  x).  58.  (30+  5)(30  +  6). 

54.  (ar  +  5x)(ar-7y).  59.  (10  +6)(10  +  2)  ;  14x16. 

55.  (a2x  +  5x)(a2x  +  3x).  60.  (20  +  3)  (20  +  4)  ;  26  X  28. 
61.  Explain  how  Types  I,  II,  III  may  be  regarded  as  spe- 
cial cases  of  Type  IV. 

136.   Type  V.     The  Square  of  a  Polynomial. 

a  +  5-f-c 

a  +  b +c 

a2  +    ab  -h    ac 

a&  +  62+      be 

ac         +     bc  +  <? 

a2  +  2  a&  +  2  ac  +  b2  +  2  be  +  c2 

Rearranging  for  convenience, 

(a  +  &  +  c)2  =  a2  +  62  +  c2  +  2a&+2ac  +  2fc:. 

Similarly,       (a  +  b  -  c)2  =  a2  +  ft2  +  c2  +  2  a&  -  2  ac  -  2  6c. 

In  words : 

The  square  of  a  polynomial  equals  the  sum  of  the  squares  of  its 
terms  plus  twice  all  products  formed  by  multiplying  each  term  by  each 
succeeding  term. 

It  is  to  be  observed  that  the  squares  of  the  terms  will  all  be 
positive  numbers  (why?),  but  that  the  double  products  may  be 
positive  or  negative  according  to  the  requirements  of  the  law 
of  signs  in  multiplication. 

Examples 

1.  (a  +  &  +  2)2  =  a2+62  +  4-f  2a6  +  4a  +  46. 

2.  (2a-&  +  l)2  =  4a2+62  +  l-4a&  +  4a-2&. 

3.  (a?  _  x  _|_  2)2  =  x4  +  x2  +  4  —  2  x*  +  4  x2  —  4  x 

=  a4  —  2xi-\-5x2  —  4  x  +  4. 


86  Multiplication 

EXERCISE 

137.  Multiply  by  Type  V: 

1.  (x  +  y  +  zf.  5.  (x*  +  x  +  iy.  9.  (m2  +  M-»2)V 

2.  (a  +  6-1)2.  (    6.  (3  a +2  6-1)2.  10>  (2^+3  w+4r)2. 

3.  (z2  +  y2  -  z2)2.  7#  (tf  +  tf^iy.  n.  (a2  -  2  ao  +  62;2. 

4.  (a&  +  bc  +  ac)2.  8.  (aj  —  xy  +  2)2.  12.  (a  +  b  +  c+a")2. 

13.  (m  +  n-p-q)\  17.  (a  -  6  +  2  c  +  d)2. 

14.  (a-;?/-z-w)2.  18.  (3a-2  6+5c)2. 

15.  (10  m  +  5  »i  +  6)2.  19.  (xy  —  yz  —  zx)*. 

16.  (2m  +  3n-4g+p)2.  20.  (a&  -  6c  4-  cd  -  da)2. 

138.  Types  VI  and  VII.     The  Cube  of  a  Binomial. 

By  actual  multiplication  we  can  find  the  value  of  (a  -f  6)3  to 
be  a3  H-  3  a26  +  3  a62  +  63.  Let  the  student  perform  the  mul- 
tiplication. 

This  gives  us  Type  VI :  (a  +  by  =  a3  +  3  a2b  +  3  aft2  +  ft3. 

Similarly,  Type  VII :     (a  -  b)3  =  a3  -  3  a2b  +  3  aft2  -  ft3. 
In  words : 

The  cube  of  the  sum  of  two  numbers  equals  the  cube  of  the  first 
plus  three  times  the  square  of  the  first  multiplied  by  the  second,  plus 
three  times  the  first  multiplied  by  the  square  of  the  second  plus  the  cube 
of  the  second. 

Let  the  student  make  the  corresponding  statement  for  Type 
VII. 

Examples 

1.  (3  x  +  2  yf  =  (3  xf  +  3(3  x)\2  y)  +  3(3  x) (2  yy  +  (2  yy 

=  27  x3  +  54  afy  +  36  xf-  +  8  ?/3. 

2.  (2  a2  -  6)3  =  (2  a2)3  -  3(2  a2)26  +  3(2  a2)62  -  63 

=  8a6-12a46  +  6a262-63. 

3.  (-a  +  2)3=(2-z)3  =  etc. 


Summary  of  Type  Forms  87 

4.    (_a_6)3  =  [C-a)  +  (-&)]3 

=  (_a)3+3(_a)2(-&)f3(-a)(-&)2  +  (-&)3 
=  -a3-3a2o-3a&2-&3. 

EXERCISE 

139.  Multiply  by  Types  Viand  VII: 

1.  (x  +  y)\  11.  (a  +  1)3.  21.  (be  -  a&2)3. 

2  (a  -  y)»  12.  (a  +  2)3.  22.  (6  +  5  a3)3. 

3.  (c  +  d)3.  13.  (2  a-  l)3.  23.  (9  m3  -  5p3)3. 

4.  (c-d)3.  14.  (a2  +  a)3.  24.  (10  -  a2)3. 

5.  (m  +  2)3.  15.  (-a  +  &)3.  25.  (2p2  -  3)3. 

6.  (m-3)3.  16.  (-a-26)3.  26.  (6  -  5p4)3. 

7.  (2  a  +  1)3.  17.  (2a  +  3)3.  27.  (m2  -  p*qf. 

8.  (a2  +  l)3.  18.  (ab  +  bc)\  28.  (7a2-2)3. 

9.  (a2  -  2)3.  19.  (abc  -  l)3.  29.  (ab  -  63)3. 
10.  (4  + a)3.  20.  (62c-ao2)3.  30.  (a^z  -  3a?2z)3- 

SUMMARY   OF   TYPE   FORMS 

140.  The  student  should  carefully  memorize  the  following 
type  forms  that  have  been  developed  in  this  chapter : 

I.  (a  +  6)2  =  a2  +  2a&  +  &2. 

II.  (a-6)2  =  a2-2a&+62. 

III.  (a-\-b)(a-  b)  =  a2-  ft2. 

IV.  (x  +  a)(x+b)=x*+(a  +  b)x+ab. 

V.    (a  +  b  +  c)2  =  a2  +  &2  +  c2  +  2  a&  +  2  ac  +  2  &c. 

(a  -h  &-  c)2  =  a2  +  ft2  +  c2+  2  a&-  2  ac-  2  6c. 
VI.    (a  +  a)3  =  a3  +  3a2&  +  3a62+&3. 
VII.    (a  -  6)3  =  a3  -  3  a26  +  3  aft2  -  63. 

Let  the  student  state  each  type  form  in  words. 


88  Multiplication 


REVIEW  EXERCISE 


141.    Give  the  answers  to  examples  1  to  Jfi  by  referring  each  to 
the  proper  type  form : 


1.  ( 

[ax  —  by)  (ax  +  by). 

21. 

(x  +  y-z)(x  +  y  +  z). 

2.    { 

[ax  —  3)(ax  —  4). 

22. 

(x-y-z)(x  +  y  +  z). 

3.    1 

[ax  -  3)(ax  -  3). 

23. 

(x  —  pr  +  q)2. 

4. 

[(a  +  b)(a  -  &)]*. 

24. 

(v  +  5  wu)(/v  —  5  ww). 

5.    ( 

[5dy  -3f. 

25. 

(v  -f-  5  wy)2. 

6. 

[1-2  m)2. 

26. 

(2  mn  —  Ipq)2. 

7. 

[ab  -  3  c)2. 

27. 

(a  —  b  +  x)(a  +  b  —  x). 

8. 

[a2-3x)\_-(3x-a2)']. 

28. 

(x2  +  7)(x2-8). 

9. 

(2p  +  q)(-2p  +  q). 

29. 

(7m-6#)2. 

10. 

(2a-7)(2a-9). 

30. 

(2/  +  2)(y  -  m). 

11. 

(2  a  +  6  -  3  c)2. 

31. 

(a  +  m)(o+p). 

12. 

(3  xy-  2  z)\ 

32. 

(2c-l  +  d)(2c-l-d) 

13. 

(3xy-2zf. 

33. 

(2  c  -  1  +  c?)2. 

14. 

(3xy  +  2z)(3xy-2z). 

34. 

(2  a2  -  3  6)3. 

15. 

(4  +  3  a2)2. 

35. 

(r  +  2i-3)2. 

16. 

(m  —  w)(ra  +  n)(m2  -f  w2). 

36. 

(d»  -  6)2. 

17. 

Jra  —  n)  (m  -h  n)  (m2  —  w2). 

37. 

(a2a2  +  5)  (a2x2  -  3). 

18. 

(2  a  -  62)3. 

38. 

(a>  +  2y-c-2d)2. 

19. 

(a  +  6  +  3)2. 

39. 

Qf  +  a;)2- 

20. 

(3  <cy  —  4  z)(  —  3  xy  —  4  z). 

40. 

(af  +  2/)2- 

Perform  the  operations  indicated  in,  the  following,  using  type 
forms  wherever  possible  : 

41.  (a  +  b)2+(a-b)2  +  (a-b)(a  +  b). 

42.  (a?  +  x  +  l)(x-l)-(a?-x  +  l)(x  +  l). 

43.  (a  +  26-c)(a-c)-(a2  +  2a&  +  c2). 

44.  (a4  -  ^  +  a2  -  1)(*  +  1).    • 


Review  Exercise  89 

45.  (v  -f  w)(vw2  —  v*w  +  v3  —  w3). 

46.  16(3  a  -  2  b)-  5(9  a  -  7  6)-  3(o  -  4  b)-  11 6. 

47.  (a6  +  3a3  +  9)(a3-3). 

48.  (m2  +  mn  +  n2)(m  —  n). 

49.  (tf-pq  +  q2)(p  +  q\ 

50.  (a2+  &2  +  l-a&-a-&)(a  +  6  +  l). 

51.  (<x?  +  y2  +  z2  —  xy-xz  —  yz)(x  +  y  +  z). 

52.  (x  +  2/  4-  «)2  — (^  +  2/  —  ^)2  +  (x-  —  y  +  2;)2  -  (—  a;  -f-  2/  +  ^. 

53.  (a  +  b)(b  +  c)  -  (c  +  d)(d  +  a)  -  (a  +  c)(6  -  d). 

54.  5[3(a  +  2  6  -  c)+  4  (a  -  b  -  c)]-  19(a  -  b  -  c). 

55.  (x2-i/2)(2x3-4ar!?/-5^2). 

56.  (a2  -  62) (2  a  -  3  6  +  5  c)  -  6(3  a2  -  2  a&  +  3  62) . 

57.  15^  +  24 2/2-(3a  +  22/)(5a  +  6  ?/). 

58.  (a2-62)(2&-3a)  +  (a  +  &)(8  6-7a)a. 

59.  (a  +  2)2-(z-l)2-33(a-3). 

60.  (1+ 2-2^X1  + a +  2  a2). 

Hint:     (1  +  x  -  2x2)(l  +  x  +  2z2)  =  [(i  +  x)_  2  s2][(l  +  z)+2x2]. 

61.  (m  +  w+])  +  g)(m-n4-|)-g). 

62.  (m  +  2p  +  9)2-f  (7?i-7i+7)- ^)2. 

63.  (a?  +  b2-ab)(a2  +  b2  +  ab). 

64.  (16ra4  +  4mY  +  2/8)(4w2-y4). 

65.  (49  afi  +  56  a%  4-  64  #2)(7  a3  -  8  y). 

66.  (2a  +  46-5c  +  2d)2. 

67.  a^  +  l2/2  +  622  +  2)2. 

68.  (a6-3a3  +  9)(a3-3). 

69.  State  the  two  binomials  whose  product  is  (1)  p*—  q*: 

(2)  m2  +  4mn  +  4n2;       (3)  a2-b*c2;  (4)  a^  +  10a;  +  25; 

(5)  m2  +  2mri  +  w2;       (6)9^-24^  +  16^;    (7)l~ia2. 


90  Multiplication 

Solve  the  folloivivg  equations  : 

70.  (a  +  2)(a+>3)-(a+-l)2  =  9. 
Solution.  (a  +  2)(a  +  3) -(a  +  l)2  =  9. 

(a2  +  5  a  +  6)-(a2  +  2  a  +  1)=9. 

a2  +  5  a  +  6  -  a2  -  2  a  -  1  =  9. 

3a +  5  =  9. 

3a  =  4. 

a  =  f 

71.  (a?  +  l)2  =  a2  +  12. 

72.  12m2+-2m  +  l-12(m2+-l)=0. 

73.  3  +  ^2-(9  +  2/2)  +  22/  =  0. 

74.  4(z-l)+l  =  7-2(2z-3). 

75.  17(l+aj)-8(aj  +  2)  =  26. 

76.  -3(a?4-2)rf7(a?  +  l)=3. 

77.  3(m  +  l)=2(m  +  3). 

78.  7(a?— 1)— 2(a?  +  2)=aj  — 3. 

79.  5(«-2)+2(aj-f 3)=17 4-2(1-*). 

80.  A  certain  fertilizer  contains  1^  times  as  much  potash  as 
nitrogen  and  4  times  as  much  phosphoric  acid  as  nitrogen.  Find 
the  amount  of  each  element  in  130  pounds  of  fertilizer. 

81.  If  10  is  added  to  a  certain  number,  the  sum  is  three 
times  the  original  number.     Find  the  number. 

82.  One  number  is  32  greater  than  another.  When  3  is 
added  to  each  number  the  greater  is  5  times  the  smaller.  Find 
the  original  numbers. 

Solution.     Let  x  =  the  smaller  number. 
Hence  x  +  32  =  the  larger  number. 

Also  x  4*  8  as  the  smaller  number  increased  by  3, 
and  x  +  35  =  the  larger  number  increased  by  3. 
Then  x  4-  86  =  6(x  +  8),    (By  the  conditions  of  the  problem.) 
or  a; +  35  =  5* +  15.     (Why?) 
.-.  35  =  4 z  +  15.     (Why?) 
.-.  20  =  4*.  (Why?) 

/.  x  =  5,  the  smaller  number, 
and  5  +  32  =  37,  the  larger  number. 


Review  Exercise  91 

83.  If  a  certain  number  is  multiplied  by  8  and  the 
product  is  increased  by  14,  the  result  exceeds  5  times  the 
original  number  by  28.     What  is  the  number  ? 

84.  A  boy  had  twice  as  much  money  as  his  sister ;  but  after 
each  had  spent  12  cents  he  found  that  he  had  3  times  as  much 
as  his  sister.     How  much  had  each  at  first? 

85.  One  number  is  5  times  another.  If  15  is  added  to  each 
number,  the  greater  will  be  3  times  the  less.  Find  the  original 
numbers. 

86.  A  rectangle  is  3  times  as  long  as  it  is  wide.  If  both 
dimensions  are  increased  by  4  inches,  it  will  be  twice  as  long  as 
it  is  wide.     Find  its  dimensions. 

87.  A  rectangle  is  3  inches  longer  than  it  is  wide.  If  both 
dimensions  are  increased  by  3  inches  the  area  will  be  increased 
by  54  square  inches.     Find  the  dimensions. 

88.  A  box  of  candy  contained  a  certain  quantity  at  35  cents 
a  pound,  twice  as  much  at  50  cents  a  pound,  and  3  times  as 
much  at  55  cents  a  pound.  If  the  mixture  cost  $  3,  how  many 
pounds  of  each  quality  did  it  contain  ? 

Solution.  Let  x  =  the  number  of  pounds  @  35  ^. 

Hence 2x  —  the  number  of  pounds  @  50 ?, 
and  3  a;  =  the  number  of  pounds  @  55^. 
Then  35  x  +  50  .  2  x  +  55  •  3  x=  300. 
Let  the  student  complete  the  solution. 

89.  A  grocer  blended  a  certain  quantity  of  coffee  at  35  cents 
a  pound  with  twice  as  much  at  32  cents  a  pound  and  4  times  as 
much  at  25  cents  a  pound.  If  the  total  value  was  $  15.92,  find 
the  number  of  pounds  of  each  in  the  mixture. 

90.  A  certain  number  of  4^  stamps,  3  times  as  many  2^ 
stamps,  and  10  times  as  many  1^  stamps  cost  $2.00.  How 
many  of  each  were  bought  ? 


VL  DIVISION 

142.  Division  has  been  denned  as  the  process  of  finding  one 
of  two  factors  when  their  product  and  the  other  factor  are 
given.  The  product  is  the  dividend,  the  given  factor  is  the 
divisor,  and  the  factor  sought  is  the  quotient. 

143.  What  is  the  rule  for  dividing  signed  numbers  ? 
(See  §  53). 

ORAL  EXERCISE 

144.  Divide  the  following : 

1.  8-K-2);   -8  +  2;  -8-(-2). 

2.  _  18-6;   -18 -(-6);  18 -(-6). 

3.  36-h(_9);   -36  -5- (-6);  36-*-(-4). 

4.  8 +(-12);  -9  +  12;  -5  +  (-15). 

5.  -lO-s-5;  5 +(-10);  -5  +  10. 

6.  12  ft. --3;  12/+ 3;  12  an- 3. 

7.  $10  +  2;  10eZ+2;  10z  +  2. 

8.  12  yd. -3;  12 y  +  3;  12  6-3. 

9.  20  mi.  +  4 ;  20  m  +  4  ;  20  x  +  4. 

10.  3  x  6a;  18a  +  3;  126-3. 

11.  5  x7r;  35r  +  5;  27fc  +  3. 

12.  21a -7;  28p  +  2;  50s -25. 

13.  $18  +  $9;  18d  +  9d;  15d  +  5d. 

14.  26ct.-2ct.;  26c  +  2c;  18r  +  6r. 

15.  24T-4T;  24*  +  4*;  28L  +  7L. 

16.  5  x8a;  40a  +  8a;  40a  +  5. 

92 


Division  93 

17.  8  x7ft;  56k+7k-,  56k  +  8. 

18.  4x(-2o);  -8a-*-(-2a);  -8a-r4. 

19.  7x(-3a);  -  21a-s-(-  3a) ;  -21a-=-7. 

20.  -8x7<;  -56*-*-(-8);  -56*-^7*. 

21.  -2x(-5r);  10r-s-(-2);  10r-=-(-5r). 

22.  3ax(-2a);  -6a2--3a;   -6a2--(-2a). 

23.  21fc-*-(-3fc);  -8a2-2a;  -  18a  -*-(-  6«). 

145.  Integral  Algebraic  Expression.  An  algebraic  expression 
is  an  integral  algebraic  expression  if  there  are  no  literal  numbers 
in  a  denominator. 

Thus,  a2  +  2  ab,  \  x  —  *-,  a  +     are  integral  algebraic  expressions, 

and      — ,  -,  a         are  fractional  algebraic  expressions. 
3  6a       c 

146.  The  Law  of  Signs  in  Division. 

The  student  should  remember  that  in  dividing  one  number 
by  another : 

1.  The  quotient  of  two  numbers  having  like  signs  is  positive. 

2.  The  quotient  of  two  numbers  having  unlike  signs  is  negative. 

147.  The  Law  of  Exponents  in  Division. 

Since  a3  •  a2  =  a5,  therefore  a5  -s-  a2  =  a3  or  a5-2, 
and  a5  -f-  a3  =  a2  or  a*-3. 
Similarly  •.•  a8  •  a3  =  a11,  .*.  a11  -5-  a8  =  a3  or  a11-8. 

and  a11  -s-  a3  =  a8  or  au~3. 
In  general,  am  -j-  a"  =  a*"'". 

The  equation' a™  -rfl»  =  a^-",  gives  in  algebraic  symbols,  the 
laiv  of  exponents  in  division. 

In  words,  this  law  may  be  stated : 

The  exponent  of  any  base  in  the  quotient  is  equal  to  its  exponent  in 
the  dividend  minus  its  exponent  in  the  divisor. 


94  Division 

Examples 

1.  a7  +  a5=:  a7"5  or  a2.  4.    —  27  -=-  24  =  —  23.  (Why  ?) 

2.  a2  -=-  -  a  =  -  a.     (Why  ?)      5.    a3*  -3-  a*  =  a2*.     (Why  ?) 

3.  -  A;6  -=-  -  fc2  =  k\  (Why  ?)     6.    ar+2  -f-  ar  =  ar+2~'  =  a2. 

ORAL  EXERCISE 

148.  Find  the  quotients  : 

1.  a4  h-  a2.  6.  -  r10-*-(-  r6).  11.  c7  -s-  c6. 

2.  m6-i-ra.  7.  — /i5 -=-(- /*).  12.  —  d8^_(_^p 

3.  a**-s-(-afy  8.  -W  +  -W>.  13.  (abf  +  ab. 

4.  2/7-f-y4.  9.  68--(-65).  14.  57-^54. 

5.  —  J96  -7-  p4.  10.  -  t9  -j-  £7.  15.  —  28  -j-  27. 

16.  (2a)8-(2a)6.  20.  m*+3-*-m*. 

17.  (a  +  6)6-s-(aH-6)8.  21.  ra*+3  ■*■  m*. 

18.  as2"  -*-  as".  22.  3n+1  -+■  3. 

19.  xbr-r-a?r.  23.  ak  +  al. 

24.    am+n  -h  aw-w. 

DIVISION  OF  MONOMIALS 

149.  State    the   definition    of    division.     Define   dividend, 
divisor,  and  quotient.     (§§51,  142.) 

8a2&3c-=-(  —  2a?b)=  —  4&2c  is  an  immediate  consequence  of 
the  definition  of  division  since  (—2  a2b)  x  (—  4  b2c)  =  8  a2b3c. 

ORAL  EXERCISE 

150.  {Jswmj  only  the  definition  of  division  give  answers  to  the 
following  and,  explain  : 

1.  21-=-  7.  6.  aW* -s- aW*. 

2.  a5 -=-<*.  7.  6p7q5  +  2pY> 

3.  3  aft-*- a.  8.  —  18  a2?/2z5  ■+•  9  xyz. 

4.  5  a^%!  -^  xyz.  9.  42  a&2  -s-  (—  7  aft). 

5.  -  7 m3w22)  -?-  mw'p.  10.  (-  33 a &2c3)--(-  11  abc). 


Division  of  Monomials  95 

151.  When  the  examples  are  simple,  the  definition  of  division 
along  with  our  previous  practice  in  multiplication  will  enable 
us  to  find  the  quotients.  A  rule  can  be  stated,  however,  that 
will  help  us  to  perform  divisions. 

To  divide  a  monomial  by  a  monomial : 

1.  Divide  the  numerical  coefficient  of  the  dividend  by  that  of  the 
divisor,  keeping  in  mind  the  law  of  signs. 

2.  Subtract  the  exponent  of  any  letter  in  the  divisor  from  the  ex- 
ponent of  that  letter  in  the  dividend  to  find  its  exponent  in  the  quotient. 

3.  Omit  from  the  quotient  any  letter  whose  exponent  in  the  dividend 
is  the  same  as  its  exponent  in  the  divisor  and  write  unchanged  in  the 
quotient  any  letter  that  occurs  only  in  the  dividend. 

Examples 

1.  -  28  abb<?  -r-  7  abc*  =  -  4  a*c. 

Why  is  the  sign  of  the  quotient  negative  ?  How  is  the 
literal  part  of  the  answer  obtained  ? 

2.  -  15  <*d*f  -*- (-  5  crPf)  =  3  c7.     Explain. 

3.  2n+4  -7-  2n~2  =  2n+4_(n_2)  =  26  =  64. 

4.  -  Sxh(a  -  b)4  --  x\a  -  b)=-3x*(a  -  b)\ 

EXERCISE 

152.  Find  the  quotients  : 

1.  15 x?y2  ■+-  3 x2y.  11.    xmyn  -t-(—  xy). 

2.  3  axy  +  2  ay.  12.    —  5  axbm~l  -r-  ( —  15  asbl~m). 
3.-7  m3n  -j-  3  mn.  3  atoc 

4.  27a2&5c3-(-9a?>c).  13'    15 a 6c' 

5.  ^a*bc*+(-ia*bc).  ^     _2a26c 

6.  J|»y  +  4«^.  4a26c 

15. 


8.  —  ax9  -7-  3  ax6.  —  2  xyz 

9.  7aWc+(-Zabc).  33 ^y 
10.    5040 a«6»-4-  720 ab2.                 '    (-3x)(-y) 


96 


Division 


Find  the  quotients : 

17  -4a5.(-7q26 

(-  2)2 .  aW 

18  ahW(-aby 

-  a4b4     * 


19. 


20. 


21. 


23. 


24. 


3»+2an+8 


2i2g7n42 
210a2 

(-<*)(- yx-c)« 

(_a)2(_6)3(_c)4- 
(_a2)(_a)2- 

-3(-a6)2 


32a3 

-34 

5e 

33(- 

5)2* 

2»+2  . 

3n+3 

25. 

2»  .  3^+2 

26.   3  aa2?/  •  6  a?/3  -h  ( — 9  alxhf). 

Write  the  work  for  this  example 
as  follows  : 


22.    ~  v  v  ~  ™)  m  3  ax*y  ■  6  ay9  =  18  a^y4 

18  a% V  -r  ( -  9  aWy2)  =  ? 

27.  4  aty  -r-  2  a*/  +  17  afy2  -h  2  a%2. 

28.  ^(a-fcy-f-a^a-fr)4. 

29.  4a^(8-  ra)3-=-4a?>(8-m). 

30.  126(?/-z)5-[-4&(?/-z)4]. 

31.  15m2(x2-l)4--3m(ic2-l)3. 

32.  15  cd4  -=-  5  <&  +  22  c5d8  X  cd  -=-  ll^d8. 

33.  10  a26  X  a&2  --  5  ab  -  5  a463  -*•  (  -  5  a26). 

34.  am  -r-  an.  37.    o2n  h-  an. 

35.  2am-5-aB.  38.    33 as+263  -r-  3 asb. 

36.  am+n-5-an.  39.   28a8+263  -s-(—  4a26). 


DIVISION    OF   A    POLYNOMIAL   BY   A   MONOMIAL 

153.    1.    Since  2x(a  +  6)  =  2a  +  26,  therefore 

(2a  +  26)-*-2  =  a  +  &, 

by  the  definition  of  division. 

%.  Since  a  (#  +  y)  =  asc  +  ay,  therefore  (ax  +  ay)  -r-  a  =  x  -\-  y. 


Division  of  a  Polynomial  by  a  Monomial       97 


3.  (2x  +  ±y)+2  =  x  +  2y. 
Check.    2{x  +  2y)  =  2x  +  ±y. 

4.  (4a2  +  10a)-=-a  =  4a  +  10. 

5.  (5a  -  106 +  15c)-*-(- 5)=- a +  26 -3c. 

ORAL  EXERCISE 


154.    Find  the  quotients : 

1.  (cy  +  dy)+y. 

2.  Q>g  +  rg)-*-g. 

3.  (cd-6a")-J-d. 

4.  (4a-8  6)-K4. 

5.  (a2  +  2a)-=-a. 

6.  (a?  +  a)+a. 

7.  (15p  +  20g)-5. 

8.  (18  ft- 27fc)-s- (-9). 

9.  (21x*-Ux)  +  7x. 

10.  (-18m2- 24  m)-?-6m. 


11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 


ax  +  bx  +  cx)-i-x. 
dx  —  dy  —  dz)  -s-  (—  d). 
2a3-6a;2  +  4a;)-^2z. 

9p>+12p)  +  (-3p). 
a26  +  a62)-f-a6. 
2a2-8a  +  10)-(-2). 
a6c-2  62c)-r-(-6c). 
14a-166+18c)-(-2). 
an  +  a2n)  -j-  an. 
an+1  —  an)  -s-  aw. 


155.   From  these  examples  we  derive  the  following  rule : 

To  divide  a  polynomial  by  a  monomial,  divide  each  term  of  the  poly- 
nomial by  the  monomial  and  unite  the  results  with  their  respective  signs. 

Examples 

2.    (15 a2  -  5  a  +  5)-*-(-  5). 
-5)15a«-5a  +  5 
-3a2  +  a-l 


1.    (3a2-6a)-?-3a. 
3a)3a2-6a 


a  — 2 

Check.     3a(a  -  2)=  3a2-  6 a. 

3.    (3  a363  -  12  a262  -  3  ab)  +  (-  3 ab). 

-  3a6)3a363-12a262-3a6 
—  a262  +  4  a6  +  1 

The  simplest  verification  of  such  exercises  is  by  using  the 
relation,  divisor  x  quotient  =  dividend,     (d  X  q  =  D.) 


98  Division 

EXERCISE 

156.    Find  the  quotients : 

1.  4  a3  -  6  a2b  -  12  ab2) -- (-  2 a). 

2.  (2^-8afy  +  10^2)H-2a;. 

3.  (21  a&c  -  35  bed  -  42  acd)  --(-7  c). 

4.  (15  a26c  -  27  ab2c  -  33  a&c2) -=-(-  3 abc). 

5.  (17a26  -13rt62)-=-2a7>. 

6.  (21az2  +  15aV)--7«a;. 

7.  (3  m2  +  4  wwi  -  9  ?i2)  ■+•(—  3). 

g     14a-76  +  7  5p-5g 

7  '  5 

9     apq-3bpq+pq  m(a  +  b)-2  m(2  a  -b) 

—  2pq  m 

12.  D  =  S  x2y  +  5xy2,  d  =  2  xy,  find  q. 

13.  q  =  a  +  &  -  3,  d  =  -  2  ab,  find  D. 

14.  D  =  2  a2cc2  —  6  asc,  ti  =  2  ax,  find  g. 

15.  Z>  =  15  a2  -  9  a5  +  18  a9,  d  =  3  a2,  find  g. 

16.  (8  ahb  -  24  a4b3  +  16  a768)  -s-  (  -  8  a4&). 

17.  (25  a?x2  +  60  a2x*  -  25  or?/2)  ■#-(-  5 »> 

18.  (21  a3 -14  a2 -a) -(-a). 

19.  (36  x*y2  -  24  ayz  -  18  xy2z)  -*> (-  6  ax/2). 

20.  (36  a10  -  24  a6  +  21  a5)  -5-  (  -  6  a5). 

21.  (100  a2bc  -  75  ab2c  +  50  a&c2) *•(-  25  a&c). 

22.  (35  c2xy  +  42  ex2  -  56  cxy)  -+-  7  ex. 

23.  (12  an+*  -  15  an+2  -  27  a"+1)  -3a. 

24.  (12  <r+3  -  15  an+2  -  27  an+1)-(-3aw). 

25.  (12  an+3  -  15  a"+2  -  27  an+l)  -«- (-  3  a"+1). 

26.  Show  that  (5"+3  +  5"+2  +  5"+1)-=-  5"  =  155. 

27.  Cato+*&r+8  -  ^  a2*+l6*+1)  +  (-  ato&*). 

28.  (22"-13  —  22n+2)  -T-  22n+1. 


Division  of  a  Polynomial  by  a  Polynomial      99 


m+n 


29.  (a^+yb3m+n  —  a'ix+2vb'lm+'n)  -r-  a?*+vb2 

30.  [(s  +  y)a+(x  +  y)&]-i-(a>  +  y). 

Solution.        *  +  *K*±jQ«±I*+jrt& 
a  +  6 

31.  [(a  +  &)a>+(a  +  &)y]-s-(a+&). 

32.  [?*2(m  +  n)  —  2  r(ra  +  »)  -f  (m  +  n)]  -5-  (ra  +  n). 

33.  [12  z2(«  +  6)3  -  32  o?y(a  +  &)2] ■*-[— 4  x(a  +  &)*]. 

34.  [2  m2(a  -  ?/2)3  -  3  m(x  -  y2)2  -  (x  -  y2) ]  -5-  (x  -  y2). 

35.  [  -  8  a2b(x  -  y)2  +  9  ab2(x  -  y)]^  a6(a?  -  2/). 

36.  [>5(a2  +  62)  -  2  z2(a2  +  62)]  +x\a2  +  62). 

37.  [12  &(a2  -  ?/2)  -  15  b2(x2  -  ?/2)]  h-  3  b(x*  -  y2). 

DIVISION  OF  A  POLYNOMIAL  BY  A  POLYNOMIAL 

157.  This  kind  of  division  will  be  understood  best  by  study- 
ing an  example 

Divide  2  a3  -  7  a2  +  10  a;  -  8  by  a:  -  2. 

1.  2a*»-7a;2-l-l0a;-8|a;-2 

2.  2  x2(x  -  2)=  2  a? -4  a2  |  2  a?2  -  3  x  +  4 

3.  _  3  a?  +  10  a;  -  8 

4.  _3a(a-2)=      -  3  x2  +  6  a; 

5.  4  x  -  8 

6.  4(z-2)=  4a;-  8 

1.    Both  dividend  and  divisor  are  arranged  in  descending  powers  of  x. 

St.  The  first  term  of  the  dividend  is  divided  by  the  first  term  of  the 
divisor  to  obtain  the  first  term  of  the  quotient,  2  x2.  The  entire  divisor 
is  then  multiplied  by  the  first  term  of  the  quotient. 

3.  The  product  obtained  is  subtracted  from  the  dividend. 

4.  The  first  term  of  the  remainder  is  divided  by  the  first  term  of  the 
divisor,  to  obtain  the  next  term  of  the  quotient,  —  3  x.  The  entire  divisor 
is  then  multiplied  by  this  second  term  of  the  quotient. 

5.  The  product  is  subtracted  from  the  last  remainder. 

6.  The  process  described  in  the  last  two  steps  is  repeated  until,  in 
exact  division,  a  remainder  zero  is  obtained. 


100  Division 

158.  The  explanation  just  given  may  be  regarded  as  a  rule 
for  the  division  of  a  polynomial  by  a  polynomial.  It  is  of  the 
greatest  importance  that  a  proper  arrangement  of  the  terms  of 
the  polynomials  be  made  at  the  beginning  and  that  the  same 
arrangement  be  observed  in  all  the  remainders  obtained  in  the 
course  of  the  work. 

Let  the  student  explain  how  the  next  term  of  the  quotient 
is  obtained.  Also  explain  all  the  operations  involved  in  steps 
5  and  6. 

To  check  examples  in  long  division  the  relation  d  •  q  =  D 
may  be  used,  or  arbitrary  values  of  the  letters  may  be  substi- 
tuted. If  the  latter  method  is  employed,  values  of  the  letters 
should  be  chosen  which  will  not  make  the  divisor  0. 

Examples 
1.   Divide  (x2  +  Sx  -  4)  by  (x  -  1). 
Division 


a2  +  3a;-4 

x2  —     x 


x  —  1  Check  :  When  x  =  2,  D  =  6, 


x  +  4  d  =  1,  q  =  6,  and  6  -=-  1  =  6. 


4»  —  4 
4sc  —  4 


2.  Divide  8 a3  +  27 63  by  4a2-6a6  +  962. 


8a3  +  27  63 
8a3-12a2&  +  18a&2 


4a2-6a&+96* 


2a  +  36 


12a26-18a&2  +  27&3 
12a25-18a62  +  27  63 

Check.    Multiply  the  divisor  by  the  quotient. 

EXERCISE 
159.   Find  the  quotients  : 

1.  (<*-5»  +  6)-!-(*_2).  4.   (a2-62)-Ka-&). 

2.  (a2-8a  +  15)-j-(a-3).         5.    (a2- a6)-5-(6  -  a). 

3.  (4  62-46+l)--(26-l).         6.    (7 a  -  14) -5- (2  -  a). 


Division  of  a  Polynomial  by  a  Polynomial    101 

7.  (3a;2  _  4  a?  +  20) -r- (a;  -2). 

8.  (6a3  -  23a26  +  25a62  -  6  63)-j-(2a-  3  6). 

9.  (30ap-6  6p  +  12cp)-i-(5a-6  +  2c). 

10.  (20 ac  -Wad-  126c  +  9&d)-s-(5  a-  36). 

11.  (3  a6d  -  3  cd  +  a6c  -  c2)  -r-  (aft  -  c). 

12.  (6  a36  +  9  a62  +  3  abc  +  2  a2c  +  3  be  +  c2)-5-(3  a6  +  c). 

13.  (a^  +  a?-4arJ  +  5a;-3)-i-(l-a?  +  a2). 

14.  (27a?-8y*)-*-(3a>-2y). 

15.  (8  a363  -  c3^3)  -f-  (4  a262  +  2  a6ca"  +  c2d2). 

16.  (a2  4-  62  +  c2  +  2  ab  -  2ac  -  2  6c)-s-(a  +  6  -  c). 

17.  (5a6  +  15a5  +  5a  +  15)-(a  +  3). 

18.  (2a4-6a3  +  3a2-3a  +  l)-(a2-3a  +  l). 

19.  (42  a4  +  41  a3  -  9a2  -  9a  - 1)-(7  a2  +  8a  +  1). 

20.  (2m4 -6m3  4- 3m2-  3m-f-l)-r(m!-3m+  1). 

21.  (6a?x  -  17 a?x2  +  14:0,0? -  3x*)  +  (2  a  -  3x). 

22.  (2  x4  +  afy  -  13  afy2  -  3  a?/  +  y*)  +  (x*  -2xy-  if). 

23.  (15  a5  +  10  a46  +  4  a362  +  6  a263  -  3  a64)  -=-  (5  a3  +  3  a62). 

24.  (21  a4  -  16  a36  +  16  a262  -  5  a63  +  2  64)  +  (3  a2  -  a6  4-  62). 

25.  (20a6  -  53a7  4-  45a9  -  a8) -- (4  a2  -  5  a3). 

26.  (a6  -  5x*y-  10 bY  4-lOafy2  4-  5a;?/4- ?/5)-r- (a;2- 2a;?/ +  y*). 

27.  (a44-2aV  4- z4  -  64)-=-(a2 +  #2  +  62). 

28.  (6  a2  4-  a6  4-  7  ac  -  12  62  +  19  6c  -  5  c2)  --  (2  a  4-  3  6  -  c). 

29.  (15a;2-29an/4-122/2-222/z-602!2)-^(5a;-32/+103). 

30.  (48  x*yA  -  80  x*tf  -  8  a;?/5  +  200  afy2)  -r-  (20  a?y2  -  4  a;?/3). 

31.  (343  aW  -  64  63a*)  --  (49  oV  +  28  a6a?  4- 16  6V). 

32.  (20a^  +  32a;-51a*J-12a;2)-=-(4a;2-7a;-8). 

33.  (32  a2  +  4562  4-  60 c2  4-  76a6  4-  88ac  +  1046c) 

-(8  a +  9  6  + 10c). 

34.  (1.2  a2  +  1.17  a6  -  11.34  62)-4-  (1.5  a  4-  5.4  6). 

35.  [x2  4-  (a  4-  c)a;  4-  ac]  -f-  (a  4-  a;) . 

36.  [f-(a-b)y-ab']^(a-y). 


102  Division 

160.  Division  with  a  Remainder.  If  the  dividend  is  not  the 
product  of  the  divisor  multiplied  by  some  integral  algebraic 
expression,  we  shall  have  a  remainder. 

1.   Divide  6  x2  -  lSx  -  3  by  2  x  +  1. 
Division 


6  x2  -13a 

6x2+    3x 


2a;  +  l 


3  x  —  8,  quotient 


-16a;-3 
-  16  x  -  8 


5,  remainder. 


2.   Divide  a?  +  3  a;2  +  7  by  a;2-2a;  +  2. 


Check 

3a; -8 

2a;  +  l 

6  x2  — 16  x 

3x- 

-8 

6  a;2 -13  a;- 

-8 

5 

6a;2  -13a; -3 


arJ  +  3a;2  +  7 
aj3_2a;24-2a; 


a;2_2a;4-2 


x  -f  5,  quotient. 


5a;2_    2a;+    7 
5a;2 -10a; +10 

8  x  —    3,  remainder. 

Unless  otherwise  directed,  perform  all  divisions  in  descending  powers  of 
some  letter,  and  continue  the  division  until  the  exponent  of  the  highest 
power  of  the  letter  of  arrangement  in  a  remainder  is  less  than  that  of  the 
highest  power  of  that  letter  in  the  divisor. 

EXERCISE 
161.   Divide,  and  check  by  the  relation  a*  •  q-\-  r  =D. 

1.  (a;2-3a;  +  5)--(a;  +  l). 

2.  (4  -  3  x2  +  2  x)  -(2  +  x). 

3.  (arJ-l)--(a;2-a;  +  l). 

4.  (3a-a3  +  2)-(l-a2). 

5.  (7a2  +  6a3  +  5a-7)-;-(3a-l). 

6.  (7a2+  6  a*  +  5  a  -7)-s-(2  a2  +  3  a  +  2). 

7.  (x3-8a3-2a2a;)-r(2a-x). 


Division  of  a  Polynomial  by  a  Polynomial     103 

8.  (_  73  #2_  25  +  56x4  +  95 a: -59 a3) 

-=-(-  llaj  +  7a?-3a*+l). 

9.  (49a3-72.a«/2  +  28^)--(7a-3?/). 

10.  (4  m4  —  m2n2  +  6  mn3  —  9  w3)  -r-  (2  m2  -  mn  +  3  ?i2). 

11.  »»-f-(a?  —  1). 

12.  For  what  value  of  k  is  x2  —  3  x  +  A:  exactly  divisible  by 

05+1? 

Solution,     x2  —  3x  +  k  \x  +  1 
s2  +    a       _  laj  — 4 
—  4x  +  & 
-4s-4 

&  +  4  =  the  remainder. 
The  division  is  exact  if  the  remainder  is  0,  or  if  k  +  4  =  0,  that  is,  if 
Jfc  =  -4. 

13.  Determine  k  so  that  ar}+3arJ  +  2.T  +  fc  shall  be  exactly 
divisible  by  x  —  2. 

14.  Determine  k  so  that  a  + 1  shall  be  an  exact  divisor  of 
a^  +  A:. 

15.  For  what  value  of  k  is  x  —  1  an  exact  divisor  of  x*  +  A;  ? 


VIL  SIMPLE  EQUATIONS 

162.  An  equation  is  a  statement  expressing  the  equality  of 
two  numbers.     (Review  §§  12-16.) 

There  are  two  essentially  different  kinds  of  algebraic  equa- 
tions as  illustrated  by  the  following : 

1.  (z  +  2)(a;-2)=z2-4. 

2.  x  +  2  =  5. 

Equation  1  is  true  for  all  values  of  x ;  equation  2  is  satisfied 
when  x  equals  3,  and  not  otherwise. 

163.  Identity.  An  equation  that  is  true  for  all  values  of 
the  letters  involved  is  an  identical  equation,  or  simply  an 
identity. 

The  symbol  =  is  sometimes  used  to  indicate  an  identity. 

The  most  frequent  use  of  the  identical  equation  is  to  indi- 
cate the  result  of  some  operation  performed  upon  algebraic 
expressions. 

The  following  are  examples  of  identical  equations : 

2  x  +  3  x  +  5  x  =  10  x. 

(x  +  3y=x2  +  6x  +  9. 
(a-b)(a  +  b)  =  a2-62. 
6m  (m  —  n)   =6m2-6w»n. 

In  the  identical  equation,  if  the  indicated  operations  are 
performed  and  the  like  terms  are  collected  in  each  member, 
the  two  members  will  be  exactly  alike. 

164.  Conditional  Equation.  An  equation  that  is  true  for 
only  certain  values  of  the  letters  involved  is  a  conditional  equa- 
tion or  simply  an  equation. 

104 


Simple  Equations  105 

A  conditional  equation  expresses  a  relation  between  an  un- 
known number  and  certain  known  numbers.  The  problem 
suggested  by  a  conditional  equation  is  that  of  rinding  for  what 
value  of  the  unknown  number  the  relation  expressed  in  the 
equation  is  true. 

The  following  are  examples  of  conditional  equations : 

2x  —  7  =  x  +  3.        True  when  x  =  10,  and  not  otherwise. 
3a  +  7  =  4a  +  7.     True  when  a  =  0  and  not  otherwise. 

2  ax  =  4  a2.  True  when  x  =  2  a  and  not  otherwise. 

ORAL  EXERCISE 

165.  1.  Is  x  +  1  =  2  a  conditional  equation  or  an  identity? 
2#  +  3  =  7? 

2.  Is  2x  —  (x'+  1)=  x  —  1   a   conditional   equation   or  an 
identity?     (x  -  l)(x  +  1)=  x*  -  1  ?     2#-l  =  #? 

3.  State  the   four   principles   used   in   solving   equations. 
(See  §  13.) 

4.  What  is  the  root  of  an  equation  ?     (See  §  16.) 

5.  What   is   the   root   of  3  +  2  =  7?     of   #-2  =  7?     of 
2#  =  3?     ofi.r=5? 

6.  What  value  of  x  satisfies  the  equation  x  —  2  =  3  ? 

/SftoM?  £to  £^e  following  are  identities  by  reducing  the  two  mem- 
bers to  the  same  expression  : 

7.  a(x  —  y)  =  ax  —  ay. 

8.  (x  +  a)(x  +  b)=x2+(a  +  b)x  +  ab. 

9.  5y  +  3-  4?/  =  ; y  +  3.  10.    llz-(5  +  10z)=z-5. 

ySoZve  tae  following  conditional  equations : 

11.  #-3  =  2.  16.   2^-8=3. 

12.  2/4-7  =  9-  #.  17.    m?  -|-4  =-10. 

13.  2x-l=5.  18.   2n=-6. 

14.  Ia>+1=4.  19-   4x-2a  +  3=-3, 

3  20.   5n-4  =  -14. 

15.  3  #  —  4  =  5. 


106  Simple  Equations 

EXERCISE 

166.    Show  that  equations  1  to  4  are  identical  equations  by  re- 
ducing the  tivo  members  to  the  same  expression. 

1.  a(6  —  c)  +  b(a  —  c)=2ab  —  c(a  +  6). 

2.  (a  +  6  -  c)(a  +  b  +  c)  =  a2  +  2  a6  +  62  -  c2. 

3.  (a2  -  2  _  2)(a2  +  x  -2)  =  (a2  -  3  x  +  2)(«2  +  3  a  +  2). 
4-  0*  +  V)(y  +  «)(«  +  »)  +  xyz  =(x  +  y  +  2!)(ajy  +  y«  +  «»). 

ify  substituting  1,  2,  awe?  3  /or  a;  m  equations  5  to  8,  s^oiy  £ta 
eacft  is  a  conditional  equation. 

5.  2a-  5  =  a;-3.  7.    (a  -  l)(a  +  2)=a2. 

6.  (a-4)2+2  =  (a-5)2-3.     8.    8a  +  7-a  =  14. 

Solve  the  following  equations  : 
9.   x  -4 x  +  3  =  0.  10.   5j>  +  18  =  3(p  +  10)-2. 

11.  31  -7a  =  41  -8*. 

12.  5x  +  13  -2a  =  100  -20a  -18+  12  a;  -15. 

13.  16^  +  10-21^  =  45-10?; -15. 

14.  7y-9-3y  +  5  =  lly-2(3  +  2y). 

15.  -40  =  5-30a  +  35-40a. 

16.  6  =  6-  9  +  36+2. 

167.  It  is  a  common  practice  in  algebra  to  use  x,  y,  and  z  to 
represent  the  unknown  numbers  in  an  equation  and  to  use 
a,  6,  c  etc.  to  represent  numbers  that  are  regarded  as  known. 

Thus,  in  the  equation  ax  =  3  a2b.  x  is  the  unknown  number  and  the  value 
of  x  is  to  be  found  in  terms  of  the  other  letters  involved.  The  value  of  x 
is  found  by  dividing  both  members  of  the  equation  by  a,  giving  x  =  3  ab. 
The  equation  x  =  3  ab  can  be  solved  for  a  or  for  b.    Thus,  dividing 

both  members  by  3 b  gives  a  =  ~-    Solve  the  equation  for  b. 

3  b 


Simple  Equations  107 

168.  Integral  Equation.  An  equation  in  which  the  unknown 
number   does   not   occur   in   any  denominator  is   an   integral 

equation. 

2  x 

Thus,  Zx  —  4  =-x  —  5  and  ax  +  b =  Sb  are  integral  equations. 

3  a 

In  the  present  chapter  all  equations  are  integral. 

169.  Solving  Equations. 

1.  Solve  3  a; -(5 -a)  =11. 

Solution.     3x—  (5  —  x)=  11. 

3  x  —  6  +  as  =  11.  (Removing  parentheses.) 

—  5  +  4  x  =  11.  (Collecting  like  terms.) 

4  x  =  16.  (Adding  5  to  both  members.) 

x  =  4.  (Dividing  both  members  by  4.) 
Let  the  student  check  the  answer  by  putting  4  for  x  in  the  original 
equation. 

4  x  =  16  is  the  simplified  form  of  the  equation  and  the  work 
done  to  reduce  the  equation  to  this  form  is  called  simplifying 
the  equation. 

2.  Solve  (x  -3)(x-2)  =  (x-  4)2. 

Solution.  (x  -  3)(x  -  2)  =  (x  -  4)2. 

x2  -  5  x  +  6  =  x2  -  8  x  +  16.  (Multiplying.) 

-bx  +  6=-  8x  +  16.  (Why?) 

8z-5a;  +  6  =  16.  (Why?) 

3x=10.  (Why?) 

x  =  V0-  (Why?) 

Let  the  student  check  the  answer  as  in  example  1. 

What  is  the  simplified  form  of  this  equation  ? 

170.  Simple  Equation.  An  equation  that  can  be  reduced  to 
an  integral  form  containing  the  first  power  of  the  unknown 
number  and  no  higher  power  is  a  simple  equation. 

Thus,  5x  —  2(3x  —  1)  =  1  is  a  simple  equation.  Also  x{x  —  5)  =  (x  —  3) 
(x  —  7)  is  a  simple  equation,  for  it  reduces  to  5  x  =  21. 

Simple  equations  are  frequently  called  first  degree  equations, 
also  linear  equations. 


108  Simple  Equations 

171.  The  type  form  of  the  simplified  equation  of  the  first 
degree  is  ax  =  b.  By  this  we  mean  that  x,  with  any  coefficient 
it  may  have  (represented  in  the  type  form  by  a),  constitutes 
the  first  member  of  the  equation,  and  the  known  term  or  terms, 
represented  by  b,  constitute  the  second  member. 

Thus,    3  x  =  7  is  in  the  form  ax  =  b ;  here  a  =  3,  and  6  =  7. 

172.  The  steps  required  to  reduce  an  equation  to  the  form 
ax—b  are  illustrated  in  the  solution  of  examples  1  and  2  of 
§  169.     The  principles  involved  are  those  stated  in  §  15. 


EXERCISE 
173.   Solve  the  first  20  equations  orally. 


1.   2  x  =  15. 

8. 

x  +  3  =  -,  7. 

15.    —  x  —  7  =-8. 

2.   a; -2  =  0. 

9. 

3  x  +  6  =  9. 

16.    7  =  2+a. 

3.     -25  =  2. 

10. 

_  x  -  7  =  0. 

17.    —5  a;  =.5. 

4.   3a  =  0. 

11. 

-2a  +  8  =  0. 

18.    ax  =  2a2. 

5.    —  x  +  3  =  0. 

12. 

7  *  -  .56. 

19.    bx  —  2  ab  =  0. 

6.    x  +  1  =  5. 

13. 

Ax  =  2. 

20.    —  a»  -f  7  a  =  0. 

7.   2a  +  l  =  5. 

14. 

5  x  -  7  =  13. 

21.   ia-3  =  2. 

22.   a- 3=2- 

-  X. 

27.    .5  a: 

-.05=  .2. 

23.    x  —  5  —  5  - 

■  X. 

28.    -100=20+(2x-25). 

24.    -2x-f  = 

-0. 

29.    .82 

x  -f-  .1  x  =  .3  sc. 

25.    17  -(16-! 

r)=l. 

30.    2x 

-  7  =  4  -  2  x. 

26.    -(18  +  a)  = 

=  19. 

31.    ±x 

_(7-2z)=4a  +  5. 

32.  6a  +  4  =  3(a  +  3)+  2(3  -  a). 

33.  5(2  x  -3)-  2(3  -2x)+  2  =  0. 

34.  2a-(4a-7)-3(5-7a;)=100. 

35.  7a-[3-2(a;-5)+3]=2. 

36.  5  -2[2  a  -(3  -10  x) -4]=  -225. 


Simple  Equations  109 

174.  It  is  sometimes  desired  to  solve  equations  when  some 
of  the  numbers  that  are  regarded  as  known  numbers  are  repre- 
sented by  letters.  The  method  is  the  same  as  that  used  in  the 
equations  already  solved. 

Example.     Solve  3  ax  —  a(2  c  +  x)  =  2  ab  —  4  ac. 

Solution.  3  ax  —  a  (2  c  4-  x)  =  2  ab  —  4  ac. 

3  ax  —  2  ac  —  ax  =  2  ab  —  4  ac. 

2  ax  —  2  ac  =  2  ab  —  4  ac. 

2  ax  =  2  aft  —  2  ac. 

x  =  b  —  c. 

EXERCISE 

175.  #o£i;e  the  following  equations,  regarding  the  last  letters  of 
the  alphabet  as  the  unknown  numbers  : 

1.  3  nx  —  nx  =  2  n2  —  4  n.  4.    bx  —  2ba  =  b2. 

2.  4  c#  —  5  c  =  ex  +  c.    ,  5.    ex  —  cd  -\-  c2  =  c3. 

3.  —  a*  +  7  a  =  a2.  6.    3  x  —  (4  a  +  7)  =  2  a  +  a. 

7.  5a-3(2c-4d)=2d  +  4c. 

8.  3a«-  2(<fo  -  a2)  =  3  a2  -  2  (to. 

9.  (a  -  4)z  =  a2  -  8  a  +  16. 

10.  5  ax -[2  ax  -  (a2  —  ax)']  =  5  a2  —  10  a. 

11.  4 a -(2  a  -36-*)=  3 a  +  6 a-  9 6. 

12.  2  a*  =  3  a2  —  a(a  -f-  *). 

13.  4a  +  3-5a  =  a-2. 

14.  a*  —  3  ab  =  2  a*  +  7  a&. 

15.  (a  -  1)*  =  a2  -  1. 

16.  3(5a-a)-2(4a-5a)=0. 

17.  12 (y  -f  n)  =  45  -  3(y  -f-  w). 

18.  5-(»  +  15)+c  =  8-&  +  2c. 

19.  2  a6«  -  a2&  =  a(bz  +  a&). 

20.  a(b  —  x)+b(c  —  x)=b{a  —  x)+bc. 

21.  If  the  same  term  occurs  in  both  members  of  an  equa- 
tion, it  may  be  dropped  from  both  members.  Why?  Would 
the  statement  be  true  if  the  terms  were  preceded  by  opposite 
signs  ? 


110  Simple  Equations 

176.  The  known  and  the  unknown  numbers  are  generally 
found  distributed  through  the  two  .members  of  an  equation. 
In  order  to  solve  an  equation,  it  is  necessary  to  reduce  it  to 
the  form  ax  ==  b. 

Consider- the  equation  5#  +  3  =  2a;  —  5. 

Solution.     1.    5  x  +  3    =2x  —  5. 

2.  E>x  =  2x—  5-3.         (Subtracting  3  from  both 

members. ) 

3.  5  x  —  2x  =  —  5  —  3.  (Subtracting  2  x  from  both 

members.) 

4.  3  x  —  —  8.  (Collecting  terms  gives  type 


form  ax  =  b. ) 


x  =  — 


If  we  compare  equation  3  with  equation  1,  we  shall  see  that  the  term 
containing  x  in  the  second  member  of  equation  1  appears  in  the  first  mem- 
ber of  equation  3  with  its  sign  changed;  also  that  the  term  not  containing 
K,  that  was  in  the  first  member,  is  now  in  the  second  member  with  its 
sign  changed. 

177.  Transposing  Terms.  Any  term  of  an  equation  may  be 
changed  from  one  member  of  the  equation  to  the  other  by 
changing  its  sign.     This  process  is  known  as  transposing  terms. 

In  writing  the  solution  of  equations  the  student  may  omit 
step  2  as  given  in  §  176,  and  write  equation  3  immediately  from 
equation  1,  describing  the  process,  as  " transposiiig  all  terms 
containing  the  unknown  number  to  the  first  member  and  all  terms 
not  containing  the  unknown  number  to  the  second  member." 

The  mechanical  process  of  transposing  is  a  simple  one,  but 
great  care  must  be  taken  not  to  lose  sight  of  the  principles  which 
underlie  the  process. 

178.  The  rule  for  solving  linear  equations  will  now  be 
stated  more  fully. 

To  solve  linear  equations  : 

1.  Perform  all  indicated  operations,  removing  all  parentheses  in  both 
members. 


Simple  Equations  111 

2.  Transpose  so  that  all  terms  containing  the  unknown  shall  be  in 
the  first  member  and  all  terms  not  containing  the  unknown  shall  be  in 
the  second  member  of  the  equation. 

3.  Collect  the  terms. 

4.  Divide  both  members  by  the  coefficient  of  the  unknown. 

Examples 

1.  Solve  Ix  -8  =  4  -(2  -  10a>). 

Solution.  7x  -  8  =4  -(2  -  10 x). 

7  x  —  8  =  4  —  2  +  10  x.  (Removing  parenthesis. ) 

7  x  —  lOx  =  4  -  2  -f  8.  (Transposing  terms.) 

—  3  x  =  10.  (Collecting  terms. ) 

x  =  -  3£.  (Dividing  by  -  3.) 

Check.  Substitute  x  =  —  3|  in  both  members  of  the  original  equa- 
tion, 7x-8  =  4-(2-10x). 

Thus,  7(-  8*)-  8  =  4  -  [2  -  10(-  8*)]. 

-  ¥  =  -  ¥• 

x  =  —  3£  satisfies  the  equation. 

2.  Solve  (x  -  5)(x  -7)=(x-  4)2  -  1. 

Solution.         (x-  5)(»-  7)  =  (x  -  4)2-  1. 

a?  _  12  X  +  35  =  x2  -  8  x  +  16  -  1.  (Why  ?) 

-12x  +  8x=  1(5  -35-1.  (Why?) 

-4x=-20.  (Why?) 

x  =  5.  (Why  ?) 

Let  the  student  check  the  result. 

3.  Solve   x(x-6)  =  (x-3)(x-2)-6. 

Solution.  x(x  -  6)  =  (x  -  3)(x  -  2)-  6. 

x2-6x  =  x2- 5x+6-6.  (Why?) 

-6x+5x  =  0.  (Why?) 

-  x  =  0.  ■  (Why  ?) 
x  =  0.  (Why  ?) 

Let  the  student  check  the  result. 

EXERCISE 
179.    Solve  the  following  equations: 

1.  3x-5  =  x  +  2.  3.    (5a-3)-(6a  +  8)=0. 

2.  4y-7  =  12y  +  2.  4.    _[»-(2-3ar)]=14. 


112  Simple  Equations 

Solve  : 
5.   7x  -25  =  15(21  -3 z)+24.       6.   4-(2a>-7)=3-4(4-5a>). 

7.  m(m  —  5)  =  (m  + 2)2  +  5. 

8.  4(a>  +  2) -2(a;  +  l)- 3(7 -»)=<>. 

9.  (p-4)(p  +  4)-(p  +  2)(p  +  3)  =  -23. 

Solution.       p2  -  16  -(p2  +  5p  +  6)  =  -  23.  (Why?) 

^•2_  16-^2-5^ -6  =-23.  (Why?) 

-5p  =-23+  16  +  6.       (Why?) 
_5j9  =  -l.  (Why?) 

P  =  *.  (Why  ?) 

Note  .  In  simplifying  the  product  -  (  p  +  2 )  ( p  +  3) ,  it  is  better  to  per- 
form the  multiplication  first  —  (j?2  +  5 p  +  6)  and  remove  the  parenthesis 
afterward,  as  the  sign  —  affects  the  whole  result  of  the  multiplication. 

10.  (v  +  2)2  - (y  +-  l)2  =(v-  2){v  -  1) -  v\ 

11.  (2a>  -  l)(3a>  +  l)-(6x-  12) («  +  3)=  0. 

12.  (4aj-7)(9a>-48)=12(3a  +  l)(a>-6). 

13.  (2  6-5)(2  6  +  5)-(4  6-ll)(6  +  l)=0. 

14.  (8z  +  5)(2a;+-7)-(4a-3)(4a;+-3)=0. 

15.  (2x-  l)(lUx  +  5)-  26a  =(8 a?  +  1)36  x  +  11. 

16.  (*  +  4)2  -  *(*  +  6)  =  22. 

17.  (2xy  +  5x(x  +  7)=(3xy  +  70. 

18.  (x  +  l)2  +(»  -  2)2  -(x  -  l)(x  +  5)-  &  =  0. 

19.  (g  -  1)2  +  (g  -  3)2  -  2(q  -  7)(q  +  15)  =  0. 

20.  ^(.t  -  1)0  +  7) -  0  +  l)(x  +  2)(»  +  3)  =  0. 

21.  (z+-l)3  =  z3  +  10  +  3z(z+2). 

22.  xv_i_l7=ii(3v  +  1). 

Solution.  £  *  -  J  -  \ |  =  ^(3  •  +  1). 

*«-i-tf  =  tt"  +  tt-     (Why?) 
*«-}*«  =  **  +  *+*!•     (Why?) 
iv  =  |.     (Why?) 
o  =  f  +  \  =  6. 
Let  the  student  check  the  answer. 

23.  \x  —  \x  +  \x  —  \x=  13. 


Simple  Equations  113 

24.  |(2  +  5aj)=i(9aj  +  2). 

25.  30(m-2)  +  im=-L(5m  +  l)+30. 

o  Id 

26.  |(5x  +  l)-K4^  +  5)=i(3a;-1)-A(6^  +  4). 

27.  |(5a?-l)-8  =  -J(4 »- 2). 

28.  i(l-aO-K2_a>)=i(3+*). 

29.  .2(o?  -  .3) -(a;  -  .l)2  =  #(.25  -  x)  +  .005. 

Solution.  .2(x  -  .3)  -  (*  -  .l)2  =  x(.25  -  x)  +  .005. 

(.2x-.06)-(x2-  .2 a;  +  .01)  =  (.25x-x2)+-005.     (Why?) 
.2sc-.06-x2  +  .2x-  .01  =  .25x-x2  +  .005.     (Why?) 
.2z  +  .2x-  .25 x  =  .06  +  .01  +  .005.     (Why?) 
.15  a;  =  .076.     (Why?) 
x  =  .5. 
Let  the  student  check  the  answer. 

30.  .25(4  c  -  6)  -  .4(5  c  -  7)  =  0. 

31.  5 a- 1.7  =  5.4a;  +  0.8. 

32.  0.123  d  +  0.138  =  0.876  -  0.123  d. 

33.  7.5  x-  2.5  -1.5  x  =  4.5. 

34.  .45  2/  -  .75  =  -  .125  -  .45  y. 

35.  .7(.8  x  -  3)-  .39  =  1.11  a  -  3(.2  a?  -  .5). 

36.  .02^  =  2(.6  -  .046/)  -  .2(.5 gr  -  2). 

37.  2fa+ia;=2i+5i. 

38.  0  =  .75&-2&-  .6fc  +  5&-9. 

39.  |(7  x  -  10)  -  |(50  - x) m  20. 

40.  5  =  3v  +  |(v+3)-i(llv-37). 

41.  |(3aj-5)-l  =  Kll-2*)  +  «. 

42.  l- 3(7^+^+7(1*- |)+t*= a 

43.  3  -(.3  -  .07d)  +  .5(.ld  +  1)  =  4  -  .2(7  -  .3d). 

44.  (1  +  6  a;)2  +(2  +  8a;)2 -(1  +  10 a?)2  =  0. 

45.  9(2  x  -  7)2  +(4  x  -  27)2  =  13(4  a;  +  15)(aj  +  6). 

46.  9[7(5  +  {3*-2}-4)-6]-8  =  l. 


114  Simple  Equations 

Solve : 

47.  3[3(3  +  J3  7i-2j-2)-2]-2  =  l. 

48.  2  +  2(2a;+3)+3(a;  +  2)=12-(a;  +  l). 

49.  i(27c  +  3)-23  =  l(6k-5). 

50.  .25(2 x  +  1)+  .2(3  x  -  1)  =  J(7»  -  1)-  x. 

51.  (8-9)(*  +  9)  =  (s+  6)24-s. 

52.  0  =  4(10- 2a;)- 3(a;- 5). 

53.  0  =  3(9-2$-  5(2 g- 9). 

54.  7(4a;-3)+3(7-8a;)=l. 

55.  8(3  i-  2)-  7t-  5(12  -3t)=  131. 

56.  7(3a;  -  6) -f- 5(a;  -  3) -4(a;- 17)  =11. 

57.  6aj-7(ll-aj)+ll=4aj-3(20-aj). 

58.  44  a? .-  32  =  84  +  31.5  a;  +  4.2  a;  +  16.8. 

EXERCISE 

180.  The  equation  x  —  15  =  24  states  in  algebraic  language 
that  some  number  diminished  by  15  is  equal  to  24,  or  that 
some  number  is  15  greater  than  24. 

In  the  same  way  translate  into  verbal  language  each  of  the 
following  algebraic  equations : 

1.  x  +  10  =  25.  4.   2x-15  =  3x. 

2.  x  +  296=  5a.  5.   25 -5a;  =  11. 

3.  3?/- 14  =40.  6.    72  -  2a;  =  9a;. 

Write  in  algebraic  language,  that  is,  make  algebraic  equations 
for  the  following  : 

7.  36  is  greater  than  some  unknown  number  by  10. 

8.  Three  times  a  number  is  14  less  than  5  times  the  number. 

9.  25  is  divided  into  two  parts  the  larger  of  which  is  n. 
What  is  the  smaller  part  ?  Make  an  equation  that  indicates 
that  3  times  one  of  the  parts  is  equal  to  2  times  the  other  part. 


Simple  Equations 


115 


10.  786  diminished  by  two  times  an  unknown  number 
is  270. 

11.  If  296  is  added  to  an  unknown  number,  the  sum  is  5 
times  the  number. 

12.  A  boy  is  three  times  as  old  as  his  sister  and  the  sum  of 
their  ages  is  16  years. 

Let  x  =  the  number  of  years  in  the  sister's  age. 

13.  If  7  is  taken  from  5  times  a  number,  the  remainder 
is  53. 

14.  If  42  is  added  to  7  times  an  unknown  number,  the  sum 
is  54. 

15.  A  rectangle  whose  length  is  a  feet,  and  whose  width  is  3 
feet  less  than  its  length,  has  an  area  of  54  square  feet. 

16.  60  is  divided  into  two  parts,  the  smaller  of  which  is  §  of 
the  larger. 

17.  50  is  divided  into  two  parts  such  that  40  %  of  one  part 
is  equal  to  20  %  of  the  other  part. 

18.  A  house  and  lot  are  together  worth  $  8500.  The  value 
of  the  house  exceeds  3  times  the  value  of  the  lot  by  $  1000. 
Find  the  value  of  each. 

19.  The  frame  of  a  picture  is  3  inches 
wide.  The  picture  is  4  inches  longer 
than  it  is  wide  and  the  area  of  the  frame 
is  252  square  inches.  Find  the  dimen- 
sions of  the  picture. 


Hint.     The  area  of  the  picture  is  ic(cc  -f  4). 
The  area  of  the  picture  and  frame  is   (x  +  6)(x  +  10).     (Why?)     The 
area  of  the  frame  is  the  difference  of  these  areas. 

20.  A  square  lot  has  a  walk  around  it  that  is  6  feet  wide. 
The  surface  of  the  walk  contains  2256  square  feet.  Find  the 
length  of  a  side  of  the  square  inside  the  walk. 


116  Simple  Equations 

THE  SOLUTION  OF  PROBLEMS 

181.  A  problem  is  a  question  proposed  for  solution.  It  in- 
volves the  finding  of  one  or  more  unknown  numbers  from 
relations  stated  in  the  problem. 

182.  In  solving  problems  the  following  suggestions  will  be 
found  useful : 

1.  The  problem  should  be  carefully  read  and  the  conditions 
stated  in  the  problem  should  be  carefully  analyzed.  Before  the 
solution  is  attempted,  the  student  should  see  clearly  the  relations 
existing  between  the  unknown  number  and  the  known  numbers. 

2.  Represent  one  of  the  unknown  numbers  by  some  letter. 
If  more  than  one  unknown  number  is  involved,  represent  them 
in  terms  of  the  same  letter. 

3.  Translate  the  verbal  language  of  the  problem  into  alge- 
braic language  in  the  form  of  an  equation. 

4.  Solve  the  equation. 

5.  Check  the  result  by  testing  whether  the  number  or  num- 
bers found  by  solving  the  equation  satisfy  the  conditions  stated 
in  the  problem.  It  is  not  sufficient  to  determine  whether  these 
numbers  satisfy  the  equation  obtained,  as  an  error  might  occur 
in  forming  the  equation. 

PROBLEMS 

183.  The  following  problems  will  illustrate  the  above  sug- 
gestions : 

1.  If  to  7  times  a  given  number  12  is  added,  the  sum  is  54. 
What  is  the  number  ? 

Solution.    There  is  but  one  unknown  number  involved. 

Let  n  =  this  number. 

Then  by  the  conditions  of  the  problem,  7  n  +  12,  or  7  times  the  number 
with  12  added  =  54,  or  the  sum. 

Therefore  the  verbal  language  of  the  problem  translated  into  algebraic 
language  gives  the  following  equation  : 


The  Solution  of  Problems  117 

7  ft  +  12  =  64. 

.-.  7ft  =  54-12,     (Why?) 
or  7  n  =  42.  (Why  ?) 

.-.  »  =  6.  (Why?) 

Check.  7  x  6  +  12  =  54. 

2.  A  certain  number  is  3  times  another  number.  The  sum  of 
the  two  numbers  is  28  less  than  twice  the  larger  number. 
What  are  the  numbers  ? 

Solution.     Two  unknown  numbers  are  involved  in  this  problem. 
Let  x  =  the  smaller  number. 
Hence  3  x  —  the  larger  number. 
Then  x  +  3x  =  6x  —  28.  (By  the  conditions  of  the  problem.) 
.-.  -2 ac  =  -28.  (Why?) 

.-.  x  =  14,  the  smaller  number, 
and  3  x  =  42,  the  larger  number. 
Check.  14  +  42  =  84  -  28,  or  56  =  66. 

3.  I  bought  for  my  library  3  volumes  at  a  certain  price,  5 
volumes  at  double  the  price,  and  4  volumes  at  J  the  price.  For 
all  I  paid  $24.     How  much  did  each  volume  cost? 

Solution.     Three  unknown  numbers  are  involved  in  this  problem. 
Let  x  =  the  number  of  dollars  paid  for  one  of  the  3  volumes. 
Hence  2  x  =  the  number  of  dollars  paid  for  one  of  the  5  volumes, 
and  |  x  =  the  number  of  dollars  paid  for  one  of  the  4  volumes. 
Then  3  a;  +  10  a;  +  3  x  =  24.     (Why?) 
Let  the  student  finish  the  solution  and  check  the  answer. 

4.  The  sum  of  the  digits  of  a  number  of  two  figures  is  9. 
By  interchanging  the  digits  the  resulting  number  will  be  27 
greater  than  the  original  number.     What  is  the  number  ? 

Solution.     Two  unknown  numbers  are  involved  in  this  problem. 
Let  x  —  the  units'  digit. 
Hence  9  —  x  =  the  tens1  digit. 

Therefore  the  original  number  is  10(9  —  x)  +  x,     (Why?) 

and  the  number  with  the  digits  interchanged  is  10  x  -4-  (9  —  x). 

The  second  number  is  27  greater  than  the  original  number. 

Then  10(9  -  x)-f  x  =  lOx  +(9  -  x)-  27. 
Let  the  student  solve  and  check. 


118  Simple  Equations 

5.  If  a  certain  number  is  doubled  and  7  is  added,  the  re- 
sult is  —  1.     Find  the  number. 

6.  The  number  of  boys  in  a  certain  school  after  being 
doubled  and  further  increased  by  10  is  60.  What  was  the 
number  at  first  ? 

7.  A  man  walked  for  a  certain  number  of  hours  at  4  miles 
an  hour  and  then  for  twice  as  long  a  time  at  3  miles  an  hour, 
covering  20  miles  in  all.     How  long  did  he  walk  at  each  rate  ? 

Hint.     Let  x  =  the  number  of  hours  at  4  miles  an  hour. 
Hence  2x=  the  number  of  hours  at  3  miles  an  hour, 
and  4  x  =  the  number  of  miles  at  4  miles  an  hour. 
Let  the  student  complete  the  solution  and  check. 

8.  Find  a  number  such  that  10  times  the  number  is  14 
less  than  3  times  the  number. 

9.  Find  the  three  consecutive  numbers  whose  sum  is  15. 

10.  The  sum  of  three  consecutive  odd  numbers  is  33.  Find 
the  numbers. 

11.  The  sum  of  four  consecutive  even  numbers  is  44.  Find 
the  numbers. 

12.  Divide  $  880  between  A  and  B  so  that  A  shall  receive 
$  50  less  than  twice  as  much  as  B. 

13.  Divide  120  into  two  parts  such  that  7  times  one  part 
equals  8  times  the  other  part. 

14.  Find  a  number  such  that  15  times  the  number  is  10 
times  as  great  as  the  sum  of  the  number  and  4. 

15.  What  price  does  a  dealer  pay  for  6  dozen  lead  pencils 
if  he  sells  them  for  5  ^  each  and  makes  a  profit  of  90  ^  ? 

16.  Find  the  number  such  that,  if  you  add  3  and  multiply 
the  sum  by  5,  the  result  is  1  greater  than  if  you  add  5  and 
multiply  by  3. 

17.  A  bag  contains  an  equal  number  of  dollars,  half  dol- 
lars, quarters,  dimes,  and  nickels.  If  the  amount  contained  in 
the  bag  is  $  47.50,  how  many  coins  of  each  kind  are  there  ? 


The  Solution  of  Problems  119 

18.  If  20  is  added  to  a  number  the  result  will  be  3  times  as 
great  as  if  4  is  subtracted  from  it.     Find  the  number. 

19.  My  neighbor's  orchard  contains  8  more  trees  than  mine 
and  together  they  contain  34  trees.  How  many  trees  does 
each  orchard  contain  ? 

20.  The  sum  of  three  numbers  is  32.  The  second  exceeds 
the  smallest  by  2  and  the  largest  is  2  less  than  twice  the 
smallest.     Find  the  numbers. 

21.  The  tens'  digit  of  a  number  is  3  times  the  units'  digit 
and  the  number  exceeds  7  times  the  sum  of  its  digits  by  9. 
What  is  the  number  ?     (See  problem  4.) 

22.  Two  towns  are  60  miles  apart.  A  starts  from  one  and 
walks  3|  miles  an  hour  toward  the  other  town  until  he  meets 
B  who  has  started  from  the  other  town  at  the  same  time  and  is 
driving  an  automobile  at  16  miles  an  hour.  After  how  long 
will  they  meet  ?     How  far  will  each  have  gone  ? 

23.  If  in  the  last  problem  A  had  started  at  8  o'clock  and 
B  at  10  o'clock,  at  what  time  would  they  have  met  ? 

24.  What  time  is  it  if  the  number  of  hours  past  noon  equals 
^  of  the  number  of  hours  to  midnight  ? 

25.  A  man  left  half  his  property  to  his  wife,  one  fifth  to  his 
daughter  and  the  remainder,  $6000,  to  his  son.  How  much 
property  did  he  leave  ? 

26.  The  deposits  in  a  bank  during  2  days  amounted  to 
$  21,000.  The  deposits  on  the  second  day  were  ^  larger  than 
on  the  first  day.     Find  the  deposits  for  each  day. 

27.  The  Washington  Monument  is  73  feet  higher  than  the 
Great  Pyramid  in  Egypt  and  the  sum  of  their  heights  is 
1037  feet.     Find  the  height  of  each. 

28.  The  sum  of  the  three  angles  of  any 
triangle  is  180°.  If  in  a  right-angled  tri- 
angle (having  one  angle  90°)  one  acute 
angle  is  twice  as  large  as  the  other,  how  large  is  each  angle  ? 


120  Simple  Equations 

29.  How  large  is  each  angle  in  a  right-angled  triangle  if 
one  acute  angle  is  10°  less  than  twice  as  large  as  the  other  ? 

30.  How  large  is  each  angle  in  a  triangle  if  the  second  angle 
is  10°  larger  than  the  smallest  and  the  largest  angle  is  equal 
to  the  sum  of  the  other  two?     (See  problem  28.) 

31.  An  Iowa  produce  dealer  ships  eggs  to  the  city  of  New 
York.  The  expense  of  shipping  and  selling  the  eggs  is  §  of 
the  original  cost  of  the  eggs.  If  the  eggs  are  sold  for  25  ^  a 
dozen,  how  much  does  the  produce  dealer  pay  for  them  ? 

32.  A  retail  dealer  received  ±^  more  from  the  sale  of  a  beef 
than  he  paid  the  packer.  How  much  did  he  pay  the  packer 
if  he  received  $84.20? 

33.  The  cost  of  shipping  wheat  from  Kansas  to  Philadelphia 
was  ||-  of  the  price  paid  to  the  Kansas  farmer.  How  much  a 
bushel  did  the  farmer  receive  if  the  shipper  received  $1.17|  a 
bushel  in  Philadelphia  ? 

34.  If  the  price  of  wheat  in  Kansas  is  J  of  the  price  de- 
livered in  Liverpool,  and  the  Kansas  farmer  receives  90^  a 
bushel,  what  is  the  price  of  wheat  in  Liverpool  ? 

35.  The  perimeter  of  a  triangle  is  60  centimeters.  The 
second  side  is  twice  as  long  as  the  shortest  and  the  longest 
side  is  6  centimeters  less  than  the  sum  of  the  other  two  sides. 
Find  the  length  of  each  side. 

36.  A  rectangular  tennis  court  is  20  feet  more  than  twice 
as  long  as  it  is  wide  and  the  distance  around  the  court  is  220 
feet.     Find  the  length  and  the  width  of  the  court. 

37.  The  height  of  one  of  the  big  trees  in  California  is  43 
feet  more  than  twice  the  distance  around  it  at  a  point  six  feet 
from  the  ground.  The  sum  of  its  height  and  girth  is  466  feet. 
Find  the  height  and  the  girth. 

38.  The  largest  package  that  can  be  sent  by  parcel  post 
must   not  exceed  72  inches   in   length   and   girth    combined. 


The  Solution  of  Problems  121 

What  is  the  largest  box  with  square  ends  that  can  be  sent, 
if  the  box  is  twice  as  long  as  it  is  wide  ? 

Hint.  If  x  =  the  number  of  inches  in  width,  4  x  =  the  number  of 
inches  in  girth. 

39.  A  star  is  added  to  the  flag  of  the  United  States  for  each 
new  state.  There  is  one  bar  on  the  flag  for  each  of  the  orig- 
inal colonies.  What  is  the  number  of  states  and  of  original 
colonies  if  the  number  of  stars  is  9  more  than  three  times  the 
number  of  bars,  and  if  the  number  of  stars  plus  the  number  of 
bars  is  61  ? 

40.  The  cost  of  a  cable  message  from  New  York  to  London 
is  25  $  a  word.  The  rate  from  San  Francisco  to  London  is 
1  ^  more  than  3  times  the  difference  between  the  rates  from 
New  York  and  San  Francisco  to  London.  What  is  the  rate 
from  San  Francisco  ? 

Hint.     Let  x  =  number  of  cents  per  word  from  San  Francisco. 
'  Thena  =  3(x-26)+l.     (Why?) 

41.  The  total  railway  mileage  of  Ohio,  Indiana,  and  Illinois 
in  1911  was  approximately  28,000  miles.  The  mileage  of  Ohio 
exceeded  that  of  Indiana  by  2000  miles,  and  Illinois  had  4000 
miles  less  than  the  other  two  states  together.  Find  the 
mileage  of  each  state. 

42.  The  steel  bridge  from  New  York  to  Long  Island  is  the 
longest  single  arch  in  the  world.  The  length  of  the  arch 
exceeds  twice  the  height  of  Washington  Monument  by  7  feet, 
and  the  sum  of  the  length  of  the  arch  and  the  height  of  the 
monument  is  1672  feet.  How  high  is  the  monument  and 
how  long  is  the  bridge  ? 

43.  The  annual  precipitation  (rainfall  and  snow)  of  Michi- 
gan is  4  times  that  of  Nevada  and  is  f  as  great  as  that  of 
Washington.  The  sum  of  the  numbers  of  inches  in  the  three 
states  is  93.5  inches.  Find  the  number  of  inches  for  each 
state. 


122  Simple  Equations 

44.  A  mark  (a  German  coin)  is  worth  4  J  ^  more  than  a  franc 
(a  French  coin).  How  much  is  each  one  worth  in  our  money 
if  two  francs  and  three  marks  are  worth  $  1.10? 

45.  The  distance  from  the  earth  to  the  moon  is  about  ^  of 
the  diameter  of  the  sun,  and  the  sum  of  the  distance  to  the 
moon  and  the  diameter  of  the  sun  is  1,128,000  miles.  Find 
the  distance  to  the  moon  and  the  diameter  of  the  sun. 

46.  A  man  bought  200  acres  of  land  for  $15,200.  For 
some  of  it  he  paid  $  70  per  acre  and  for  the  rest  he  paid  $  85 
per  acre.     How  much  did  he  buy  at  each  price  ? 

Solution.  Let  x  =  the  number  of  acres  at  $  70. 

Hence  200  —  x=  the  number  of  acres  at  #  85. 

Also  70  x=  the  number  of  dollars  for  1st  part, 
and  85(200  —  x)  =  the  number  of  dollars  for  2d  part. 
Then  70  x  +  85(200  —  as)  ss  15,200.     (By  the  conditions  of  the  problem.) 
Let  the  student  complete  the  solution. 

47.  A  grocer  bought  70  pounds  of  coffee  for  $  19.20.  Part 
of  it  cost  24  ^  a  pound  and  the  rest  cost  30  ^  a  pound.  How 
many  pounds  of  each  kind  did  he  buy  ? 

48.  A  grocer  wishes  to  mix  coffee  that  he  sells  at  28  cents 
a  pound  with  other  coffee  that  he  sells  at  35  cents,  to  get  a 
blend  that  he  can  sell  at  30  cents  a  pound.  How  many 
pounds  of  each  should  he  take  to  get  70  pounds  of  the  mixture  ? 

49.  How  many  pounds  each  of  50  ^  tea  and  75  $  tea  should 
be  mixed  to  get  20  pounds  worth  60^  a  pound  ? 

50.  How  many  pounds  each  of  white  Dutch  clover  seed 
worth  40  i  a  pound  and  blue  grass  seed  worth  20^  a  pound, 
should  be  used  to  make  100  pounds  of  a  lawn  grass  mixture 
worth  25^  a  pound  ? 

51.  A  man  loaned  $  2000,  part  at  6  %  and  part  at  4  %.  The 
interest  on  each  part  was  the  same.  How  much  was  loaned  at 
each  rate  ? 

52.  A  man  loaned  %  1000,  part  at  5  %  and  part  at  6  %.  His 
interest  was  $  57.     How  much  was  loaned  at  each  rate  ? 


Rules  and  Formulas  123 

RULES  AND  FORMULAS 

184.  A  rule  can  often  be  more  easily  remembered  if  ex- 
pressed in  algebraic  language  by  means  of  a  formula. 

Thus,  i  =  prt  (where  i  =  the  interest,  p  =  the  principal,  r  =  the  rate, 
and  t  =  the  time  in  years)  is  a  formula  by  means  of  which  the  interest  on 
a  sum  of  money  can  be  found  when  the  principal,  the  rate,  and  the  time 
are  given. 

A  —  -  bh  is  a  formula  by  means  of  which 

■ 

the  area.  A,  of  a  triangle  can  be  found  when 
the  base,  b,  and  the  altitude,  h,  are  given. 

185.  Translating  Rules  into  Formulas.  & 

The  area  of  a  trapezoid  equals  the  sum  of  the  parallel  sides 
multiplied  by  i  the  altitude.     The  formula 

A^lhCb  +  bz) 


is  a  short  way  of  writing  the  rule,  where 
A  represents  the  area,  h  the  altitude, 
and  bx  and  b2  (read  b  sub  one  and  b  sub 
two)  represent  the  two  parallel  sides  of 
Tl  '    the  trapezoid. 

ORAL   EXERCISE 
186.    Express  each  of  the  following  rules  as  a  formula : 

1.  The  area,  A,  of  a  circle  is  equal  to  -n-  (3.1416)  times  the 
square  of  the  radius,  r. 

2.  The  area,  A,  of  a  rectangle  is  equal  to  the  product  of 
its  two  dimensions,  a  and  b. 

3.  The  diagonal,  d,  of  a  square  is  equal  to  one  of  its  sides, 
s,  multiplied  by  V2. 

4.  The  distance,  d,  that  a  train  goes  is  equal  to  the  product 
of  the  rate  per  hour,  r,  multiplied  by  the  number  of  hours  (t). 

5.  The  profit,  p,  is  equal  to  the  selling  price,  s,  minus  the 
cost,  c. 

6.  The  rate  per  cent  of  profit,  r,  is  equal  to  the  quotient  of 
the  selling  price,  s,  minus  the  cost,  c,  divided  by  the  cost. 


/\ 

c 

/' 

//b 

124  Simple  Equations 

187.  Translating  Formulas  into  Rules.  The  formula  for  find- 
ing the  area  of  a  rectangle  is  A  =  ab,  where  A  is  the  area,  and 
a  and  6  are  its  two  dimensions.  Hence  this  formula  is  an 
abbreviation  for  the  rule  :  The  area  of  a  rectangle  is  equal  to  the 
product  of  its  two  dimensions. 

EXERCISE 

188.  Express  each  of  the  following  formulas  as  rules : 

1.  C  =  2  irr,  where  C  represents  the  circumference  of  a  circle 
and  r  its  radius. 

2.  V=  a>b  -  c,  a  formula  for  the  volume  of  a  rectangular 
parallelepiped  whose  dimensions  are  a,  6, 
and  c. 

3.  rt=d,  d-^r—t  and  d-i-t=r,  where 
d,  r,  and  t  represent  distance,  rate,  and 
time  respectively. 

4.  A  =  irr2,  where  A  represents  the  area 
of  a  circle  and  r  its  radius. 

5.  V=\  hirr2  where  V=  volume  of  a  cone,  h  =  altitude  and 
r  =  radius  of  base. 

189.  The  Use  of  Formulas.  In  using  a  formula  the  problem 
may  be  merely  that  of  evaluating  an  expression,  or  it  may 
involve  the  solution  of  an  equation. 

1.  Find  the  area  of  a  triangle  whose  base  is  6  inches  long 
and  whose  altitude  is  5  inches. 

Solution.  Substituting  6  and  5  for  b  and  h  respectively  in  the  formula 
A  =  lbh,  we  have  ^1  =  | .  6  •  5  =  15. 

.*.  the  area  of  the  triangle  is  15  sq.  in. 

2.  Find  the  altitude  of  a  triangle  whose  area  is  20  square 
inches  and  whose  base  is  10  inches  long. 

Solution.  20  =  \  .  10  •  h  or  20  =  5  h. 

The  altitude  of  the  triangle  is  4  inches. 


Rules  and  Formulas  125 

EXERCISE 

190.  1.  Find  the  area  of  a  triangle  whose  base  is  9  inches 
and  whose  altitude  is  7  inches. 

2.  Find  the  base  of  a  triangle  whose  altitude  is  5  inches 
and  whose  area  is  18  square  inches. 

In  problems  in  simple  interest,  if  p  represents  the  principal, 
r  the  rate  of  interest,  t  the  time  expressed  in  years,  %  the 
interest,  and  a  the  amount  (principal  plus  interest),  we  have 
the  following  formulas : 

(1)  i  =  prt,    (2)  a  =  p+ior  a=p  +  prt 

3.  What  is  the  interest  on  $  900  at  6  %  for  2  years  ? 

4.  What  principal  will  produce  $288  interest  in  4  years 
at  6  %  ? 

Hint  :  Substituting  in  i  =  prt,  288  =  p  x  .06  x  4,  or  288  =  .24p.     Solve. 

5.  How  long  will  it  take  $  1200  at  5  %  to  produce  $  210  ? 

6.  At  what  rate  will  $  1800  produce  $  252  interest  in  2 
years  ? 

7.  What  principal  will  amount  to  $  1220  in  4  years  at  5^f0  ? 
Hint.     Substituting  in  a  =  p  +prt,  1220  =p  -f  .22 p.     Solve. 

8.  What  principal  at  5  %  will  yield  an  annual  income  of 
$350? 

9.  Using  the  formula  of  §  185  find  the  area  of  a  trapezoid 
whose  parallel  sides  are  12  inches  and  15  inches  and  whose 
altitude  is  8  inches. 

10.  Using  the  same  formula,  find  the  altitude  of  a  trapezoid 
knowing  that  the  parallel  sides  are  8  feet  and  4  feet  long  and 
that  the  area  is  50  square  feet. 

Hint.  Substitute  the  numbers  given  for  the  proper  letters  of  the 
formula  and  solve  the  resulting  equation  for  h. 

11.  In  a  trapezoid  ^1=72  square  inches,  6X  «=  17  inches, 
h  =  2|  inches  ;  find  b2. 


126 


Simple  Equations 


To  find  the  perimeter  of  a  rectangle,  we  have  the  formula : 

12.  Find  p,  when  a  =  12  inches,  and  b  =  7  inches. 

13.  Find  a  when  p  =  12  feet  and  b  =  20  feet. 

A  formula  for  the  approximate 

length,  /,  of  an  open  belt  pass- 
ing around  two  pulleys,  as  in  the 
figure,  is  given  by  the  equation 
/  =  2  d  +  3^(7?  +  r),  where  d  is  the 

distance  between  the  centers  of  the  pulleys  and  R  and  r  are 

the  radii  of  the  pulleys. 

14.  Find  the  length  of  the  belt  when  the  pulleys  have  radii 
of  2  feet  and  1  foot  respectively,  and  the  distance  between 
their  centers  is  7  feet. 

15.  Find  d,  when  I  =  27 §  feet,  R  =  2  feet,  r  =  11  feet. 

16.  Find  R  if  /  =  80.5  feet,  d  =  24  feet,  r  =  4  feet. 

17.  Z  =  12  feet  3  inches,  R  =  10  inches,  r  =  6  inches,  find  d. 

The  formula  for  a  crossed 
belt  is 

l=2d  +  3l(R  +  r). 

18.  Find  the  length  of  a 
crossed  belt  when  the  centers  of 
the  pulleys  are  8  feet  apart  and  their  radii  are  1.5  feet  and  .9  foot. 

19.  Find  d  if  the  length  of  the  crossed  belt  is  26  feet  and 
the  radii  of  the  pulleys  are  1.8  feet  and  1.2  feet. 


REVIEW  EXERCISE 

191.    1.    Define  algebraic  expression,  monomial,  polynomial, 
binomial,  trinomial.     Illustrate  each. 

2.    Define  and  contrast  factor  and  term  ;  degree  and  power ; 


exponent  and  coefficient. 


Review  Exercise  127 

3.  What  are  the  four  principles  used  in  solving  equations  ? 

4.  What  is  the  base  in  (-3)2?  in  —  32?  Compare  the 
values  of  32  and  (-  3)2 ;  of  33  and  (-  3)3 ;  (-  3)2  and  -  32. 

5.  If  a  series  of  numbers  are  connected  by  the  signs  4- ,  — , 
x ,  -r- ,  in  what  order  must  the  operations  be  performed  ? 
3  +  2-5-  3(-  3)=? 

6.  Give  the  rules  for  adding  algebraic  expressions.  How 
can  results  be  tested? 

7.  What  kind  of  expression  is  obtained  by  adding  two  like 
monomials  ?     two  unlike  monomials  ? 

8.  What  is  the  rule  for  subtraction?  How  may  results  be 
tested  ? 

9.  Give  the  rule  for  finding  the  sign  of  a  product.  What 
sign  has  the  product  in  (—  l)3  •  (—  2)5  •  (—  7)? 

10.  State  the  law  of  exponents  in  multiplication.  Without 
using  the  law  of  exponents  explain  why  a2  •  a3  =  a5. 

11.  Give  the  rules  for  multiplying,  (a)  two  monomials ; 
(b)  a  polynomial  by  a  monomial ;  (c)  two  polynomials.  How 
can  you  test  the  correctness  of  the  product  ? 

12.  Give  the  rules  for  dividing,  (a)  one  monomial  by  an- 
other ;  (6)  a  polynomial  by  a  monomial ;  (c)  one  polynomial 
by  another.     How  can  you  test  the  correctness  of  the  quotient? 

13.  In  multiplication,  two  factors  are  given  and  the  product 
is  required.  In  division,  which  two  of  these  three  numbers  are 
given  and  which  is  required  ? 

14.  How  may  the  subtrahend  and  the  difference  be  com- 
bined to  get  the  minuend  ?  How  may  the  divisor  and  the 
quotient  be  combined  to  get  the  dividend  ? 

15.  If  the  minuend  is  positive  and  the  subtrahend  is  nega- 
tive, what  is  the  sign  of  the  difference  ? 

16.  What  is  the  sign  of  the  sum  of  two  negative  numbers  ? 
of  the  difference  ?  of  the  product  ?  of  the  quotient  ? 


128  Simple  Equations 

17.  If  the  product  and  the  multiplier  have  the  same  sign, 
what  is  the  sign  of  the  multiplicand  ?  If  they  have  opposite 
signs,  what  is  the  sign  of  the  multiplicand  ? 

18.  If  the  dividend  and  the  divisor  have  the  same  sign,  what 
is  the  sign  of  the  quotient  ?  If  they  have  opposite  signs,  what 
is  the  sign  of  the  quotient  ? 

19.  In  the  expression  a  —  3  m  +  4p  —  7  b  +n  —  15,  inclose 
the  third  and  the  fourth  terms  in  a  parenthesis  preceded  by  the 
minus  sign,  then  inclose  this  parenthesis  and  the  term  imme- 
diately preceding  and  the  one  immediately  following  it  in 
brackets  preceded  by  the  minus  sign,  leaving  the  final  expres- 
sion of  the  same  value  as  the  original  polynomial. 

20.  How  can  you  test  the  correctness  of  the  factors  of  an 
algebraic  expression  ? 

21.  What  is  the  difference  in  meaning  between  Sx  and  sc3? 
Illustrate  when  x  =  5. 

22.  What  is  the  difference  in  meaning  between  the  square 
of  the  difference  of  two  numbers  and  the  difference  of  the 
squares  of  the  same  numbers  ?  Illustrate  when  the  numbers 
are  a  and  b. 

23.  Why  do  we  change  signs  when  removing  a  parenthesis 
preceded  by  the  minus  sign  ? 

24.  Give  the  rule  for  squaring  a  binomial. 

25.  When  is  a  binomial  the  product  of  the  sum  and  the 
difference  of  two  numbers  ? 

26.  What  must  be  added  to  x2  4-  4  x  to  make  it  an  exact 
square  ?     What  must  be  added  to  x2  +  6  x  -f  4  ? 

27.  Is  the  product  changed  if  an  even  number  of  its  factors 
have  their  signs  changed  ?  Compare  the  value  of  (a  —  b)2  with 
(b-ay. 

28.  State  the  rule  for  cubing  a  binomial. 

29.  Define  equation ;  identical  equation ;  conditional  equa- 
tion. 


Review  Exercise  129 

30.  What  is  the  root  of  an  equation  ?  Are  any  of  the  num- 
bers, 1,  2,  -  3,  5  roots  of  x2  -  2x  -  15  =  0? 

31.  Describe  briefly  the  steps  used  in  solving  an  equation. 
What  is  meant  by  transposing  ?  What  principles  are  used  in 
transposing  ? 

32.  What  important  difference  is  there  between  the  equa- 
tions (x  —  l)(x  +  l)=o;2-l  and  x2  —  1  =  0. 

33.  Subtract  1  —  x  4-  2  x2  from  x3.  Subtract  the  same  ex- 
pression from  0. 

34.  Divide  x2  —  7  x  -f  k  by  x  —  2,  giving  quotient  and  re- 
mainder. How  long  should  such  divisions  be  continued? 
For  what  value  of  k  will  this  division  be  exact  ? 

In  examples  35  to  45,  A  =  a2  +  3  ab  -  4  b2,  B  —  az  +  4  a2b 
-a&2-463,  C=a  +  4b,  D  =  a3  +  64  63. 

Perform  the  indicated  operations : 

35.  B-(Ab  +  D).  37.    B  -  AC. 

36.  D+C.  38.    Aa  -  D  +  Cb\ 

39.  The  minuend  is  B  and  the  difference  is  D ;  find  the  sub- 
trahend. 

40.  The  divisor  is  C,  the  quotient  is  A,  and  the  remainder 
is  16  b3 ;  find  the  dividend. 

41.  Find  the  value  of  B  when  a  =  —  2  and  b  =  —  3. 

42.  Find  the  value  of  B  when  a  =  b. 

43.  Multiply  B  by  C  and  verify  the  result  by  using  a  =  1, 
6=2. 

44.  J.  is  quotient,  C  is  divisor ;  find  dividend. 

45.  C2  =  ?      Z>2  =  ? 

46.  Expand  by  type  forms  : 

(a)  (3  s3 -4  a)2,  (d)  (4a*  +  5#2)2. 

(6)   (a>  +  2</)3.  (e)    (4a?y-3a»)« 

(c)  (8  -  25  a»)(8  +  25 a3).  (/)  (36  +  4  x)2. 


130  Simple  Equations 

47.  (a)  12a*  +  2±ai  =  2a  (?). 

(6)  9  ra2  +  16p2  -  24  wip  =  (3  m  -  4jp)  (?). 

(c)  8  af»  —  12  xhj  +  6  ^y2  -  f  =  i  ?  -  ?)3. 

(d)  9  x4^4  -  36  mhi*  =  (3  x7y2  -  6mw)  (?). 

48.  Simplify  2  n(a  -  /i)2  -  (a2  -  3  ah)a  -  (a  -  h)(a  -  3  fc)a. 

49.  Simplify  (8  a?  -  12  afy  +■  6  xf-f)^(2  x-yy  +  (y-2  x). 

50.  Divide  [3 a:2(a*  +  a)2+(x  +  a)]  by  (x  +  a). 

51.  If  you  add  to  a  number  i  of  it  and  7,  the  result  is  27. 
What  is  the  number  ? 

tfoZve  the  following  equations : 

52.  3.5  x  +  9.3  =  1.25  +  10.3. 

53.  25  -6(3  x  -15)=  5. 

54.  3-3(2a?H-4)=6-4(2aj  +  3). 

55.  2(a--3)-3(l  -2 a?)  =3(2  -  a?) - 2(5 - 3 x). 

56.  (tf  +  5)2-(a;2  +  95)=0. 

57.  (6 a;  +  4) (8 x  -  5)-(4 a;  +  12)(12  x -  21)=  0. 

58.  (a>  +  12) (a)  -  12)  -(a?  +  8)2  =  0. 

59.  Using  x  as  the  unknown  number,  write  equations  whose 
solutions  will  answer  the  following  questions  : 

(a)  What  number  added  to  23.7  gives  14.81  ? 

(b)  What  number  subtracted  from  12.84  gives  14.81  ? 

(c)  What  number  multiplied  by  98  gives  12.25  ? 

(d)  What  number  multiplied  by  -l  gives  12.25? 

(e)  To  what  number  must  |  be  added  if  the  result  is  to  be 
equal  to  that  obtained  by  multiplying  the  number  by  f  ? 

60.  Solve  the  equations  of  59. 

61.  Divide  a4  -  bA  by  a  —  b. 

62.  What   must  be  added   to  a?4  —  3  a?  —  x  +  5  to  produce 
x3  —  x  —  1  ? 

63.  Solve  (4z-l)(a;  +  3)-4a32-(-10a;  +  3)  +  6  =  0. 


Review  Exercise  131 

64.  (a5  -  48  -  17  a8  +  52  a  +  12c**)-s-(a  -  2  +  a2)=  ? 

65.  -bind  (#  -f  l)3  —  (x  —  l)3  when  #  =  —  |. 

66.  Simplify  a  -  [3  a  -  b  -  2(6  -  a)  +  3(a  -  2  6)]. 

67.  Divide  a,*3  by  x  +  1. 

68.  State  in  algebraic  symbols  the  type  forms  of  multiplica- 
tion given  as  special  products. 

69.  Find  the  quotients  : 

(a)  [3x  +  3y  +  a(x  +  y)-]+(x  +  y). 

(b)  (l-9a2e6)-r-(l-  3  at3). 

(c)  (l-9^  +  8^y)-(l-8^). 

70.  Electric  light  bills  are  paid  at  the  rate  of  14  ^  each  for 
the  first  few  units  used  and  4^  each  for  the  remainder.  A  bill 
for  35  units  was  $2.  How  many  units  at  each  price  were 
paid  for  ? 

71.  Think  of  a  number,  double  it,  add  13,  subtract  5,  divide 
by  2.  Show  that  the  final  result  will  always  be  4  greater 
than  the  number  you  first  thought  of. 

72.  Think  of  a  number,  multiply  it  by  3,  add  6,  divide  by  3, 
subtract  the  original  number.  Show  that  the  result  will  always 
be  2. 

73.  Divide  x3  —  10  x  +  17  by  x  —  a  until  the  remainder  does 
not  contain  x.     Compare  the  remainder  with  the  dividend. 

74.  Divide  x3  —  5  by  x  —  a  until  the  remainder  does  not  con- 
tain x  and  compare  as  in  example  73. 

75.  Divide  x*  —  5  by  x  —  2. 


VIIL    FACTORING 

192.  If  two  or  more  algebraic  expressions  are  multiplied 
together,  the  result  is  their  product,  and  the  expressions  multi- 
plied are  factors  of  the  product. 

Thus,  3x5  =  15.  .-.3  and  5  are  factors  of  15,  also  m(x  +  y)  = 
mx  +  my.  Here  mx  +  my  is  the  product  of  which  m  and  (x  +  y)  are 
the  factors. 

Note.  Unless  otherwise  stated,  expressions  containing  fractions  or 
indicated  roots  are  not  considered  as  factors.  Thus,  although  3  =  5  x  f , 
or  V3  x  V3,  we  shall  not  in  this  chapter  consider  these  expressions  as 
factors  of  3. 

193.  Prime  Number.  A  number  which  has  no  integral  fac- 
tors except  itself  and  1  is  a  prime  number. 

Thus,  7,  23,  a  +  b,  a2  +  3  b'2  are  prime  numbers. 

Prime  numbers  used  as  factors  are  prime  factors. 
Thus,  a  and  a  +  b  are  the  prime  factors  of  a2  +  ab. 

194.  The  student  will  recall  that  division  is  the  process  of 
finding  one  of  two  factors  when  their  product  and  the  other 
factor  are  given.  In  factoring  it  is  required  to  find  both  fac- 
tors when  only  the  product  is  given.  Thus  factoring,  like 
division,  is  an  inverse  of  multiplication. 

In  arithmetic  we  learned  a  multiplication  table  and  could 
factor  all  products  that  occur  in  the  table  from  memory.  For 
example,  42  =  6  x  7. 

Corresponding  to  this  we  have  in  algebra  some  type  forms 
of  multiplication  (Chapter  V),  and  we  shall  be  able  to  factor 
the  corresponding  products  from  memory. 

132 


Factoring  133 

Thus,  &  -  y2  =  (x  +  y)(x-y), 

and  a2  +  2ab  +  b2  =  (a  +  b)2. 

Success  in  this  kind  of  factoring  depends  upon  ability  to 
recognize  these  type  products. 

ORAL  EXERCISES 
195.    Factor  the  following : 


1. 

ra2  -  n2. 

5. 

a2-4.                      9.    h2  +  2hk  +  k2. 

2. 

p2  -  q2. 

6. 

4a2  —  9.                 10.    a2  +  2a  +  l. 

3. 

&  -  d2. 

7. 

a2+  2  xy+  y2.       11.  p2-2pg4-92. 

4. 

h2  -  k\ 

8. 

m2-2mp+i>2.   12.   c2-2c-f-l. 

3. 

a2  +  4« 

+  4. 

16.    s2  —  4  s£  4- 4  £2. 

4. 

x2  +  6x 

+  9. 

17.   /i2  +  10 /*  4- 25. 

.5. 

ra2-8? 

71-1-16. 

18.   a2&2  4. 2  ab  +  1. 

196.  When  we  try  to  factor  products  not  found  in  the 
multiplication  table  in  arithmetic,  we  generally  look  for  an 
exact  divisor,  following  certain  rules  regarding  divisors. 

Thus,  195  is  clearly  divisible  by  5.  If  we  divide,  we  get  a  quotient  39. 
Therefore  195  =  5  x  39. 

Similarly/ in  an  algebraic  expression,  if  we  can  find  a  divisor, 
we  can  factor  the  expression. 

For  example,  3  a2  +  6  ab  is  clearly  divisible  by  3  a,  and  the  quotient  is 
a  +  2  6,  hence  the  factors  of  3  a2  +  6  ab  are  3  a  and  a  +  2  b. 
Also  a(x  +  y)-  b(x  +  y)  =  (x  +  y)(a  -6). 

We  proceed  to  classify  some  of  the  simpler  types  of  factoring. 

197.  Case  I.   Factors  of  Monomials.     Square  Root. 
The  factors  of  monomials  are  generally  evident. 

If  the  two  factors  of  a  product  are  equal,  either  of  them  is 

the  square  root  of  the  product.     The  radical  sign  (V     )  is  used 
to  indicate  that  the  square  root  of  a  number  is  to  be  taken. 


Thus,  V9  x*  =  VS  x2  •  3  x2  =  3  x2. 


134  Factoring 

ORAL  EXERCISE 
198.   Factor  the  following : 

1.  x*  =  x  (?).  5.   «5  =  a2  (?)  =  a4  (?). 

2.  3  x2y=xy  (?)  =  3  y  (?).  6.    cr+2  =  an  (?)  =  an+1  (?). 

3.  abb3  =  a6  (?)  =  a2b  (?).  7.    x2n  =  a"  (?)  =  #-i  (?). 

4.  72  a%2=9  a?  (?)  =  8  ay  (?).  8.   m%*  =  ra2™  (?). 

9.    6  e*+4  =  2  e3  (?)  =  2  e*+2  (?). 
10.   39  aW  =  a&c  (?)  =  13  a262c2  (?). 

Find  the  indicated  roots  : 

11.  VI.  19.    V49  a10bu  OQ       /625  a2W 

12.  Vtf.  20.    V121  mW.  A  225  TO""' 

13.  V4tf.  ai.    V169^.  27-    Vi^.4- 

14.  V9aW.  22.    Vl44a468.  28.    -v/^TT" 

. ^  a2y4 

15.  V25  ay.  23.    V^V  p4 — 

16.  V3tW.  24.     VS-  29'    Wrf&S" 

it.  vsw      25     fiy.  30     /49Z. 

18.    V32  •  52  •  x*yK  yl  n2  '     \    32 


199.    Case  II.      Type   Form  ab  +  ac,  —  Polynomials  with   the 
Same  Monomial  Factor  in  Each  Term. 

1.  Factor  4  a  4-  6  b  —  10  c. 

Solution.     4  a  +  6  b  -  10  c  =  2(2  a  +  3  b—  6  c). 

2.  Factor  2  a3?/  +  6  a;2?/2  -  8  xy3. 

Solution.     2xsy  +  6  x2y2  —  8  xy3  =  2  xy{x2  +  3  xy  —  4  y2) . 

3.  Has  x2  4-  #?/  4-  y2  a  monomial  factor  ? 

4.  How  may  results  in  factoring  be  verified  ? 


Factoring  135 

200.  To  factor  polynomials  of  the  form  ab  +  ac : 

1 .  Find  the  greatest  monomial  factor  of  every  term  of  the  polynomial. 

2.  Divide  the  polynomial  by  this  monomial. 

3.  The  factors  will  be  the  monomial  and  the  quotient  obtained  by 
dividing. 

EXERCISE 

201.  Factor: 

1.  cy  +  dy.  8.  —  5p  +  log.  15.  3a2  +  4a&. 

2.  mp+np.  9.  —7a -21  a.  16.  3x2  +  6x. 

3.  rs  +  ps.  10.  —  pr—  qr.  17.  4#2  —  8  #. 

4.  /aA;  +  mk.  11.  2  ax  -f-  6  ax.  18.  jwj  +  qx2. 

5.  2p  +  2r.  12.  5 ]>q- 10 pq.  19.  35  -  14  a2. 

6.  4s  +  8.y.  13.  Sax-6bx.  20.  42  -  28 p2. 

7.  6r-12r.  14.  -6cfc-18  67c.  21.  48  a  +  6  a2. 

22.  a3  -  3  a2  +  7  a.  30.  2  x*y  -  6  a2?/2  -  8  xty5. 

23.  3  ar»- 21  a2-  15  x.  31.  5  m6  -  2  m4n  +  10  m3?i. 

24.  27  x2  -  3  a?/  +  15  y2.  32.  a2  — afc  +  ac  — a. 

25.  ab2  +  a26  +  a262.  33.  a2  —  ay  —  6  a  +  6  xz. 

26.  ax-\-bx  —  a.  34.  ac  —  6c  +  ac2d  —  bdc. 

27.  an  +  an+1.  35.  ax—  bx  —  axy  +  #. 

28.  a3  +  a26  —  ab2.  36.  x*  —  x—  xhj  —  xy2. 

29.  x2  —  xy  —  xf.  37.  a3 +  4  a2 +  3  a. 

ORAL  EXERCISE 

202.  1.    State  in  algebraic  symbols  and  in  words  the  rule  for 
squaring  the  sum  or  the  difference  of  two  numbers.     (See  §  130.) 

Find  the  indicated  squares  : 

2.  {x  +  yf.  7.    (a-2)2.  12.  (5p-6)2. 

3.  (x-yf.  8.    (m  +  5)2.  13.  (2  a-  3  b)\ 

4.  (m  +  nf.  9.    (7  -  r)2.  14.  (-3  +  2  m)2. 

5.  (p-q)2.  10.    (0--9)2.  15.  (2m-3)2. 

6.  (h  +  k)\  11.    (2z+3)2.  16.  (_a;_2  2/)2. 


136  Factoring 

Square : 

17.  O-10w)2.  20.  -(x  +  y)\  23.  (_4m+3a2)2. 

18.  (2mn-5p)2.  21.  -(2-3  a)2.  24.  (-  m2w  -p2?)2. 

19.  (4aj»  +  5y)*.  22.  (-5a3-2  64)2.  25.  (-a2 -a)2. 

203.  If  a  trinomial  contains  tivo  terms  that  are  perfect  squares, 
and  if  the  absolute  value  of  the  other  term  is  tivice  the  product  of 
their  square  roots,  the  trinomial  is  the  square  of  a  binomial. 

Note.  To  make  a2+b2  a  perfect  square  we  add  2  ab  (twice  the  product 
of  the  square  roots  of  a2  and  b2).  To  make  a2  4-  2  ab  a  perfect  square  we 
add  b2  (the  square  of  the  quotient  2ab+2a).  To  make  16p2+2bq2 
a  perfect  square  we  add  2  x  4p  x  bq  =  40jp#.  IQp2  +  40  pq  +  26  q2 
—  (4  P  +  o  <?)2.  Also  to  make  4  A2  +  12  Aft  a  perfect  square  we  add 
[12  Aife-r(2x2A)]8  =  (3 ft)2  =  9 ft2.     4 h2  +  12  Aft  +  9 ft2  =  (2 A  +  3 k)2. 

EXERCISE 

204.  Which  of  the  following  are  squares  of  binomials  ? 

1.  a2  +  4a  +  4.  5.  m2  +  mn  +  n2. 

2.  a2 -4  a -4.  6.  2a  +  a2  +  l. 

3.  x*y2-2xy  +  l.  7.  l-6c  +  9c. 

4.  x2  —  x  -4-  ^.  8.  p2  +  2pq  —  q2. 

9.   How  many  negative  signs  may  there  be  in  the  square  of 

a  binomial  ? 

10.   Compare  the  square  of  a  —  b  with  the  square  of  b  —  a. 

Compare  the  square  of  a  —  b  with  the  square  of  a  -f-  fe- 
ll.  What  term  must  be  supplied  in  each  of  the  following  in 

order  to  make  the  trinomial  the  square  of  a  binomial  ?     Of 

what  binomial  is  the  resulting  trinomial  the  square  ? 
(a)x>+(     )  +  16.  (/)  a2b2-6abc  +  (    ). 

(6)  4a2  +  (    )+962.  fo)  16m2-(    )-f-25n2. 

(c)  25a2 -f  10a+(    ).  (ft)  36p4  +  24p2  +  (    ). 

(d)  (    )+8a  +  16.  (0  (    )+16a26  +  62. 

(e)  (    )-8z  +  4.  (/)  49 m27i2  +  (    )  +  25/>2. 


Factoring  137 

205.  Case  III.  Type  Forms  a2  +  2  ab  +  ft2  and  a2  -  2  ab  +  62, 
—  the  Square  of  a  Binomial. 

Factor  a2  4-  4  a  +  4. 

Solution.     The   first  term  is  the  square  of  a,  the  last  term  is  the 
square  of  2,  and  the  middle  term  is  twice  the  product  of  a  and  2. 
...  a2  +  4  a  +  4  =  (0  +  2)  (a  +  2)  or  (a  +  2)2. 

206.  To  factor  a  trinomial  that  is  the  square  of  a  binomial : 

1.  Arrange  the  trinomial  in  order  of  the  powers  of  some  letter. 

2.  Extract  the  square  roots  of  the  first  and  last  terms  and  connect  the 
results  with  the  sign  of  the  middle  term.  The  square  of  this  binomial 
equals  the  trinomial. 

In  algebraic  symbols  this  rule  may  be  written : 

a2  +  2ab  +  &  =  (a  +  b)2 
or  a2  -  2  ab  +  ft2  =  (a  -  b)2. 

Examples 

1.  Factor  4  x2y2  —  12  xyz  +  9  z2. 

Solution.    The  arrangement  is  in  order  of  powers  of  x. 
Step  2  of  the  rule  gives  4  x2y2  -  12  xyz  +  9z2=(2xy-3  z)2. 

2.  Factor  9^  +  4^+12^. 
Solution.     9  x*  +  12  x3  4  4  x2  =  (3  x2  +  2  x)2. 

EXERCISE 

207.  Factor  the  following : 

1.  x2  -  20a  4-  100.  9.  x2  +  12  x  +  36. 

2.  m2-f-6m  +  9.  10.  64-462  +  4. 

3.  4a2+4a  +  l.  11.  a2a2  -  12 aa  +  36. 

4.  62-106c  +  25c2.  12.  &4-462c  +  4c2. 

5.  xl  +  2x2  +  l.  13.  9a2  +  6a;  +  l. 

6.  as  _  40  a4  +  400.  14.  m2  +  14  mn*  +  49  n6. 

7.  xy  4-  4  a#  +  4.  15.  r%*  -  10  rs£  4-  25  £2. 

8.  a6  4-  6a3  4- 9.  16.  a2  +  4 c2  —  4ac. 


138  Factoring 

Factor : 

17.  2ax  +  a2+x>.  24.  81a  +  18a2+«3. 

18.  49 p2  —  28pq  +  4g2.  (Hint.     First  apply  Case  II.) 

19.  x2b2  +  a~y2  -  2  aba*/.  25.  49  a2b2c  +  28a&c  +  4  c. 

20.  1  -  20a  +  100a2.  26.  9a462+4c2d4-12a26cd2. 

21.  9- 12  a  +  4a2.  27.  8  a3  -  16  a2  +  8  a. 

22.  25  a2  +  4  a2c2- 20  a2c.  28.  aV  +  4aa;2  +  4. 

23.  81a2&4-18a62  +  l.  29.  9  x2  -  42  a  +  49. 

30.  3a2x  +  6ax2  +  3^. 

31.  m2  +  2mw  +  ?*2  +  2(ra  +  w)  +  1. 

32.  4:X2  +  £xy  +  y2  +  2(2x  +  y)z  +  z2. 

208.  State  in  algebraic  symbols  and  in  words  the  rule  for 
multiplying  the  sum  of  two  numbers  by  the  difference  of  the 
same  two  numbers.     See  §  132. 

ORAL  EXERCISE 

209.  Find  the  products  : 

1.  (x  +  2a)(x-2a).  4.    [(a  +  &)+ c][(a  +  &)- c]. 

2.  {xy-  z2)(xy  +  z2).  5.    [x  -(a  +  &)][>  +  (a  +  6)]. 

3.  (6aa-9a2)(6aa  +  9a2).      6.    (12a2  -  62)(12a2  +  62). 

Find  the  quotients  : 

7.  (x2-y2)  +  (x-y).  10.    (25  -  16a2)--(5  -  4a). 

8.  (a2_9)_j_(a+3).  n.    (a4_i6)H_(a2_4). 

9.  (9r2- l)H-(l+3r).        12.    [(a  +  &)2-c2]-=-[(a4-&)-c]. 

What  binomial  ivill  exactly  divide  each  of  the  following  ? 
What  is  the  quotient  ? 

13.  a2 -16.  15.    144  r2  -121s2. 

14.  l-4a2.  16.    a4-b\ 


Factoring  139 


Factor  the  following : 

17.   g2  -  h\ 

20.   x2  -  y2z2. 

18.    a2 -4. 

21.    1  -81  a* 

19.   4-9  c2. 

22.    x*  —  2/6. 

210.  Case  IV.      Type  Form  a2  —  b2,  —  the  Difference  of  Two 
Squares. 

211.  To  factor  the  difference  of  two  squares : 

1.  Find  the  square  roots  of  the  squares. 

2.  Use  the  sum  of  the  square  roots  for  one  factor  and  their  difference 
for  the  other  factor. 

In  algebraic  symbols  this  may  be  written : 

a*-&=(a  +  b)(a-b). 

Examples 

1.  16a2-9=(4a  +  3)(4a-3). 

2.  x4-y2=(x2  +  y)(x2-y). 

3.  (a-&)2-9c2=(a-&  +  3c)(a-6-3c). 

EXERCISE 

212.  Factor  the  following  : 

1.  9  a2 -49.  11.  25  a264-c6. 

2.  a4 -4  a2.  12.  121  x4  -  144  y\ 

3.  25a262-c4.  13.  16  a4  -  1. 

4.  a4 -49.  14.  25  a6-  16  b\ 

5.  16c4-25#.         "  15.  x4-x\ 

6.  a?y2  -  h2x\  16.  a%  —  b\     (Four  factors.) 

7.  x4-y\  17.  9  a2b4  -  25  c4d6. 

8.  9-16  a2&4.  18.  a*  -  100  afyV. 

9.  144  -81,  19.  1  -400  a;4. 
10.  64  -  x*.  20.  9  -  a2. 


140  Factoring 

Factor : 

21.  3 -27  a2.  23.    169  -  z4. 

22.  6  m2-  24.  24.    16  -  a4b\     (Three  factors.) 

213.  Case  IV,  a.     Sometimes  polynomials  of  four  or  six  terms 
can  be  written  as  the  difference  of  two  squares. 

1.  Factor  ra2  +  2  mn  +  n2  —  x2. 

Solution,     m2  +  2  mn  -f  n2  —  x2  =  (m  +  n)2  —  x2 

=  (m  +  n  +  x)  (m  +  n  —  x) . 

2.  Factor  a2-x2  +  2xy  -  y2. 

Solution.       a2  —  x2  +  2xy  —  y2  =  a2  —(x2  —  2  xy  +  y2) 

'=  (a  +  x-y)(a-x  +  y). 

3.  Factor  a2  +  2  a&  +  b2  —  c2  +  2  cd  -  d2. 

Solution,     a2  +  2  a&  +  b2  -  c2  +  2  cd  —  d2 

=  (a2  +  2  a&  +  62)-(c2  -  2  dc  +  <F) 

=  (a  +  6)2-(c-d)2. 

=  (a+6  +  c-c?)(a  +  5-c  +  d). 

Care  must  be  taken  not  to  make  mistakes  when  inserting  or 
removing  parentheses. 

EXERCISE 

214.  Factor  the  following  : 

1.  c2-2cd  +  d2-4.  7.  a4 -\- 2  a2b2  +  b4  -  a2b2. 

2.  a2b2  -  a2  +  2  ab  -  b2.  8.  x2  +  6  «  +  9  -  1. 

3.  X2  _  a2  _|_  2  ab  -  b2.  9.  2  x2  +  4  a;  +  2  -  8. 

4.  a2  -  2  a&  4-  62  -  c2.  10.  25  -  a2  -  b2  -  2  ab. 

5.  i_^_2^-2/2.  11.  2-2(a-6)2. 

6.  9-a2-62  +  2a&.  12.  3m2-6mn  +  3?i2-12. 

13.  (p-qy-(p  +  q)2. 

14.  x2  +  2/2  -  a2  —  b2  —2xy-\-2  ab. 

15.  x2  —  2  xy  +  y2  —  m2  +  2  ran  —  n2. 

16.  4z2-12ax-c2-fc2  +  9a2-2cfc. 

17.  4  a;2?/2  —  (x2  +  y2  —  z2)2.     (Four  factors.) 


Factoring  141 

215.  Case  IV,  b.  Expressions  of  the  form  a2x*  +  kx2  +  ft2,  can 
sometimes  be  factored  by  a  method  known  as  "completing  the 
square."  By  this  method  these  expressions  are  put  into  the  form 
of  the  difference  of  two  squares  and  factored  accordingly. 

1.  Factor  9  a4  +  8  a2  +  4. 

Solution.     9  a4  +  8  a2  -f  4  =  9  a4  +  12  a2  +  4  -4  a2.     (Why?) 
=  (3  a2  +  2)2  -  4  a2 
=  (3  a2  +  2  +  2  a)  (3  a2  +  2  -  2  a) 
or  (3  a2  +  2  a  +  2)(3  a2  -  2  a  +  2). 

2.  Factor  a?4  +  x2y2  +  y4. 

Suggestion,     x4  +  asfy8  +  y4  =  x4  +  2  x2y2  +  y2  —  x2y2. 
Let  the  student  complete  the  solution. 

3.  Factor  a4  +  4. 

Solution,    a4  +  4  =  a4  +  4  a2  4-  4  —  4  a2. 

=  (a2  +  2  +  2  a)  (a2  +  2  -  2  a) . 
or  (a24-2a  +  2)(a2-2a  +  2). 

216.  It  is  clear  that  factoring  by  this  method  will  depend 
upon  our  ability  to  change  the  given  expression  into  a  trinomial 
square  by  adding  a  monomial  that  is  itself  a  perfect  square. 
This  monomial  is  then  subtracted,  thus  leaving  the  original  ex- 
pression unchanged  in  value,  but  written  in  the  form  of  the 
difference  of  two  squares. 

EXERCISE 

217.  Factor  the  following  : 

1.  a4-6a2  +  l.  8.  l  +  a24-a4. 

2.  4a4  +  3a2  +  l.  9.  16  a4  +  4  a2x2  4-  x4. 

3.  as*— 3a*  +  l.  10.  a4  +  9 64  -  3 a262. 

4.  **  —  23a2 +  1.  11.  r*-14r2  +  25. 

5.  a4  —  11  a22/2  +  y4.  12.  r4  -  15  r2  +  25. 

6.  l  +  2a262  +  9a4&4.  13.  r4  +  r2  +  25. 

7.  a4  +  a%4  +  #8-  14.  4  t4  -  21  *2s2  +  s4. 


142  Factoring 

Factor : 

15.  p4  +  5p2  +  9.  22.  16  a4  -  28  a2b2  +  9  64. 

16.  p4  +  2j92  +  9.  23.  16a4  +  20a262  +  9  64. 

17.  49 a^  +  25 i/4  +  66 x*y\  24.  a4 -10a2 +  9. 

18.  49  a4  +  25 1/4- 74  a2*/2.  25.  169  -  127  62  +  9  64. 

19.  a4 +  4^.  26.  169  +  69  h2  +  9  64. 

20.  a4 +  64.  27.  a4n  +  a2n  +  1. 

21.  a4¥  +  8  a2b2c2  +  36  c4.  28.  49  a4"  +  10  a2"  +  1. 
29.  Consider  or4  —  5  a2  +  4  in  each  of  the  following  ways  : 

(a)  a4  —  5  a2  +  4  =  a4  —  4  a2 +  4  —  a2  =  etc. 
(6)  a4  —  5  a2  +  4  =  a4  +  4  a2  +  4  —  9  a2  =  etc. 
(c)  a4  -  5  a2  4-  4  =  a4  -  5  a2  +  -\5-  -  J  =  etc. 

REVIEW  EXERCISE 

218.   Factor  the  following : 

1.  a2a  +  ax2  -  a2a2.  8.  3a2  +  12  a?/  +  12y2  -  3. 

2.  4  a2  -  4  a  +  1.  9.  4  a2  -  4  ac  -f  c2. 

3.  25  a2?/2-  130  a^+169  z2.  10.  (2  a  +  l)2  -(2  x  -  l)2. 

4.  2a2  +  20a  +  50.  11.  a4  +  5 a2y2  -f  9 y4. 

5.  3m2  -3%2.  12.  a2 -16a +  64. 

6.  2  a4 +  8.  13.  a2 -144. 

7.  10  a -40  a3.  14.  81  a4-  16  b\ 

15.  25raVa8-20m2tta4?/z2  +  4  2/2z4. 

16.  1-20  a +  100  a2. 

17.  (a  +  2)2+2(a  +  2)+l. 

18.  m2  +  2 raw  +  w2  —  2 (m  +  n)p  +p2. 

19.  (a-4)2-4(a-4)+4. 

20.  (2a  +  3*/)2-(2a-3?/)2. 

21.  m2  —  n2+p2-  g2—  2m/)  +  2ng. 

22.  a16  —  616  into  five  factors. 


Factoring  143 

23.  (a2  +  9  b2  -  c2)2  -  36  a2b2  into  4  factors. 

24.  8  ra2™2  —  2  (m2  +  n2  —  p2)2. 

25.  4  a&  —  62  +  c2  —  4  a2  +  9  d2  +  6  cd. 

26.  4  r2s2  -  16  r2  +  £2  -  4  rs*. 

27.  18  (a2m  +  bm)2  -  32  (a2m  -  bmf. 

28.  27a2-90a  +  75-12fc2-12&-3. 

29.  a2n  -  1.  •  31.    a^  +  6  a8'  +  9  xr. 

30.  «2r  -  4  xr  -f  4.  32.    —  81  +  a2  +  2  ab  +  62. 

219.   Case  V.     Grouping  for  a  Polynomial  Divisor. 

1.  Given  the  expression  a  (x  +  y)  4-  b  (x  +  y). 

Solution.     It  is  clear  that  this  expression  can  be  divided  by  x  +  y. 
The  division  can  be  performed  mentally  giving  as  a.  quotient  a  +  b. 
.\  a(x  +  y)  +  b(x  +  y)  =  (x  +  y) (a  +  b). 

2.  Given  ab  —  2  br  —  2  as  +  4  rs. 

Solution.  If  we  group  the  first  two  terms  and  also  the  last  two  terms, 
and  remove  from  each  group  a  monomial  factor,  we  discover  a  binomial 
divisor. 

ab  -  2  br  —  2  as  +  4  rs  =  (ab  —  2  br)  —  (2  as  —  4  rs) 
=  b(a-2r)-2s(a-2r) 
=  (a-2r)(b-2s). 

3.  Factor  a2  —  b2  +  c(a  -  6). 

Solution.     Here  the  binomial  divisor  is  evidently  a  —  b. 
...  a2-62  +  c(a-6)  =  (a-6)(a  +  6  +  c). 

4.  What  binomial  will  divide  x2  —  y2—(x  +  y)2 ?  What  is 
the  quotient  ?     What  are  the  factors  ? 

5.  Factor  (a;  +  y)2  —  9  +  4(a?  4-  y  —  3). 

Solution,     (x  +  y)2  -  9  —  4(as  +  y  —  3) 

=  (x  +  y  +  S)(x  +  y  -  3)-  4(x  +  y  -  3) 

=  (x  +  2/-3)(x  +  y  +  8-4j 
=  (x  +  y-3)(x  +  y-l). 


144  Factoring 

220.  To  factor  by  grouping : 

Arrange  the  terms  into  groups  and  factor  each  group  separately  by 
any  of  the  preceding  methods.  If  the  same  polynomial  factor  occurs 
in  each  group,  make  it  one  of  the  factors  of  the  expression,  and  divide  by 
it  to  get  the  other  factor. 

Success  in  factoring  by  this  method  requires  great  care  in  in- 
serting and  removing  parentheses. 

The  student  is  warned  against  thinking  that  an  expression 
is  factored  when  some  group  of  its  terms  is  factored. 

Thus,  a2  —  b2  +  c2  =  (a  +  b) (a  —  b)  +  c2,  but  the  expression  a2  —  62  + 
c2,  is  not  factored  and  cannot  be  factored. 

Factoring  by  grouping  is  frequently  used  in  factoring  poly- 
nomials of  four  terms.  If  such  a  polynomial  is  the  product 
of  two  binomial  factors,  when  it  is  properly  grouped  and  the 
monomial  factors  are  separated  from  each  group,  one  of  the 
binomials  will  appear  as  an  exact  divisor  of  the  expression. 

EXERCISE 

221.  Factor  the  following  : 

1.  ab  4-  ac  4-  bd  -f-  cd.  5.   3  a  4-  3  —  pa  —  p. 

2.  a2  +  2a&  +  3ac  +  66c.  6.    xy  +  3x  +  y  +  3. 

3.  4  ab  +  4  ac  —  bd  —  cd.  7.  pq  —  pr  4-  qr  —  r2. 

4.  5  ab  —  3  b  +  5  ac  —  3  c.  8.    1  —  x  4-  xy  —  x*y. 

9.  m2n  —  p2mn  -f-  mx  —  p2x. 

10.  bx  +  by  +  bz  4-  ex  -f  cy  +  cz. 

11.  a\l  -  c)  -  b2(c  -  1). 

12.  a3x  +  ab2x  —  a2by  —  bzy. 

13.  2  ax  —  2bx  —  2  ay +  2  by. 

14.  5wv  —  5w  +  v  —  1. 

15.  (x  +  ?/)2  4- O  4- 2/). 

16.  (a  4-  b)2  4-  2(a2  -  b2).       ■ 

17.  (a-6)-2(a2-&2). 


Factoring  145 

18.  (a  +  2)(a2-9)-(a  +  2)(a  +  3)-a-3. 

19.  x2(m  —  n)  -f-  2  ax(m  —  n)  +  a2(m  —  »). 

20.  (a-26)2-4-5(a-2o  +  2). 

21.  3(x  —  y)2  —  ay  +  a#. 

22.  3(^-2/2)_(2/_a.)_h2(^  +  22/). 
Hint.     First  collect  like  terms. 

23.  28a3-12x2-  112  a +  48. 

24.  ^  +  S^p2  —  5p  —  15. 

25.  ra4  +  5  m3  —  ra2  —  5  ra. 

26.  c2  -  4  d2  +  3  ac2  +  12  acd  +  12  ad2. 
Hint.     Group  first  two  and  last  three. 

27.  (m  +  2)0i  +  3)-(w  +  3)(j>  +  2). 

28.  3(ra  -f  nf  —  5(m  +  n)2  +  m  +  w. 

29.  ri2(2m-l)-2n(2ra-l)  +  (2m -1). 

30.  p2+p  +  q+pq. 

31.  a3  +  a2-6a-6. 

32.  a#  4-  bx  +  c#  -f  a  -f  6  +  c. 

33.  ac  —  5  6c  +  a  —  5  &  —  6  c  —  6. 

34.  a2 -(a  +&)#  +  «&• 
Hint.     Remove  parenthesis. 

35.  x2  +  (a  —  b)x  —  ab. 

36.  y2-(a-2)y-2a. 

37.  a2 +(7  -y)x-  ly. 

38.  p2-(a2-a)p-a3. 

39.  4r2  +  2(d-c)r-cd. 

40.  a4+(&-4)a2-4&. 


146  Factoring 

222.  Case  VI.  Type  Form  jc2-f-  bx  +  c,  — the  Product  of  Two 
Binomials  having  a  Common  Term. 

x  +  2  x  +  p 

x  +  5  x-\-  q 

x2  -\-  2  x  x2  -f-  px 

5x+  10  qx  +  pq 

x2  +  7.x  +  10  x2+(i9  +  q)x+pq. 

It  is  readily  seen  that  x  4-  2  and  x  +  5  are  the  factors  of  se2  -f  7  a;  +  10, 
and  x  +  p  and  a;  +  q  are  the  factors  of  x2  +  (p  +  q)x  +  pq.  In  factoring 
a  trinomial  of  the  type  form  x2  +  bx  +  c  (sometimes  called  a  quadratic 
trinomial)  the  first  term  of  each  factor  is  x  and  the  sum  of  the  second 
terms  of  the  factors  is  b  and  their  product  is  c.     (See  §  134.) 

ORAL  EXERCISE 

223.  Multiply  the  following  : 

1.  (a +  3)0 +  4).  6.  (xy-&)(xy  +  7). 

2.  (a>-l)(&  +  3).  7.  (o6  +  l)(a&  +  3). 

3.  (a  +  2)(a  +  5).  8.  (m2  +  3)(m2  -  7). 

4.  (a  -  2){x  +  3).  9.  (m2-  3)(m2  +  7). 

5.  (a2  +  7){a2  -  9).  10.  (x  -f  a) (a  +  &)• 

11.  Find  two  numbers  whose  sum  is  5  and  whose  product  is 
6.     Factor  a? +  5  a; +  6. 

12.  Find  two  numbers  whose  sum  is  —  5  and  whose  product 
is  6.     Factor  x2  —  5  x  +  6. 

13.  Find  two  numbers  whose  sum  is  —  3  and  whose  product 
is  -  10.     Factor  a2  -  3  a  -  10. 

14.  Find  two  numbers  whose  product  is  —  6  and  whose  sum 
is  —  1.     Factor  x2  —  x  —  6. 

15.  Find  two  numbers  whose  product  is  6  a2  and  whose  sum 
is  5  a.     Factor  x2  +  5  ax  -f-  6  a2. 

224.  To  factor  a  trinomial  of  the  form  x2  +  bx  +  c. 

1.  Find  two  numbers  whose  product  is  c  and  whose  sum  is  b,  the 
coefficient  of  x  with  its  proper  sign. 


Factoring  147 

2.  Write  for  the  factors  two  binomials,  the  first  term  of  each  being  x 
and  the  second  terms  the  numbers  found. 

Notice  that  when  c  is  negative  the  second  terms  of  the  two 
binomial  factors  have  unlike  signs.  When  c  is  positive  the 
second  terms  of  the  factors  have  the  same  sign  as  the  middle 
term. 

Examples 

1.  Factor  x2  —  4  x  —  5. 

Solution.     The  two  factors  of  —  5  whose  sum  is  —  4  are  —  5  and  +  1. 
...  cc2_4x_5  _  (£_5)(X+  i). 

A  variation  of  this  type  which  will  not  cause  any  difficulty  is  seen  in 
the  following : 

2.  Factor  a2  +  3  ab  -  18  b2. 

Solution.  The  two  factors  of  —  18  b2  whose  sum  is  +  3  b  are  6  b 
and  —  3  b. 

...  a2  +  3  a6  -  18  b2  =  (a  +  6  b)(a  -  3  b). 

3.  Factor  a2b2  -  abc  -  20  c2. 

Solution.  The  two  factors  of  —  20  c2  whose  sum  is  —  c  are  —5c 
and  4  c. 

.-.  a2b2  -  abc  -  20  c2  =  (ab  -  5  c)(ab  +  4  c). 

EXERCISE 

225.   Factor  the  following : 

1.  x2-lx  +  Vd.        5.    a2-5a-14.      9.   2/2  +  3?/-l8. 

2.  z2-7a-30.        6.    ra2  +  5ra  +  6.     io.    b2  +  3  b  +  2. 

3.  a2 -7  a +  12.        7.    62-96  +  20.     11.    c2  -  3  cd  +  2  d2. 

4.  a2 +  5  a— 14.        8.   a2  -fa  — 42.       12.   g2  +  9g  +  20. 

13.  a2  -  11  03/  +  30  ?/2.  18.   x2  -  27  a  -f  182. 

14.  a2  -f  (m  +  n)x  +  raw.  19.    x2  —  28  a  +  195. 

15.  r2  +  ar+6r-f  a&.  20.   a2  -  29  a  +  210. 

16.  #2+(a  +  &-fe>+ac+&c.  21.    m2  -  45  m  +  164. 

17.  a2+(a+&+c>-+a&+ac.  22.    a2-4a6-12  62. 


1 48  Factoring 

Factor : 

23.  s2-3.s-18.  25.   m27*2  +  15wmp  +  50p2. 

24.  p2  —  5px  +  6x2.  26.   x2  +  2xy  —  3oy2. 

27.  t2  +  2t  +  1+50  +  1)  +  6. 

28.  7  m2  —  14  mw  +  7  w2  —  91(m  —  ri)  +  84. 

29.  3 a3 +  30  a2- 288  a.  35.  a4-4a2  +  4. 

30.  ^-^x-yV  36.  b4  +  4  62c  -  21  c2. 

31.  ap2  —  (3  g  —  2)op— 6  aq.  37.  a2n  —  a"  —  2. 

32.  r2-7rs-18s2.  38.  x2n  +  xn-2. 

33.  #4  +  4  a2  —  45.  39.  x2n  —2xn—  15. 

34.  a4  -  5  a2  +  4.  40.  a3  -  5  a2x  -  24  ax2. 

Case  VII.  Type  Form  ax2  +  bx+  c,  —  the  General  Quadratic 
Trinomial. 

226.  This  type  differs  from  the  last  type  in  that  the  coeffi- 
cient of  x2  is  not  positive  1.  The  expression  is  factored  by 
changing  the  trinomial  into  a  polynomial  of  four  terms  and 
then  grouping. 

Factor  6  x2  +  19  x  +  15. 

Solution.     6sc2+  19z  +  15  =  6jc2+  10* +  9*+  15 

=  2x(Sx  +  5)  +3(3  a; +  6) 

=  (3z  +  5)(2x  +  3). 

The  problem  here  is  to  change  the  original  trinomial  into 
the  form  in  black-faced  type.  The  numbers  10  and  9  which 
replace  19  as  the  coefficient  of  x  are  two  factors  of  90,  and  90 
is  the  product  of  6  times  15. 

227.  To  factor  a  general  quadratic  trinomial  ax2  +  bx  +  c : 

1.  Find  two  numbers  whose  product  is  a  x  c,  and  whose  sum  is  b. 

2.  Replace  bx  by  two  terms  having  these  numbers  as  the  coefficients 
of*. 

3.  Factor  by  grouping  as  in  Case  V. 


Factoring  149 

Examples 

1.  Factor  6  x2  +  7  x  -  3. 

Solution.     Here  a  =  6,  b  =  7,  c  =  -  3.     The  two  factors  of  a  .  c,  that 
is  of  6  •  (—  3)  =  —  18,  whose  sum  is  7,  are  —  2  and  9.    Then  we  write 

6x2  +  7x-3  =  6x2-2x  +  9x-3. 

=  2x(3a;-l)  +  3(3z-  1) 
=  (3x-l)(2z  +  3). 

2.  Factor  3  a2  -  11  a&  +  6  b2. 

Solution.     Two  factors  of  3  •  6  b'2  =  18  &2,  whose  sum  is  —  1 1  b  are  —  2b 
and  -  9  b. 

3  a2  -  11  ab  +  6  62  =  3  a2  -  2  ab  -  9  a&  +  6  &2 

=  a(3a-2&)-3  6(3a-2&) 

=  (3a-26)(a-36). 

EXERCISE 

228.   Factor  the  following  : 

1.  2z2  +  3a;-2.  13.   7g2-20g-3. 

2.  2x2-  7a- 15.  14.   2ra2  +  9m-5. 

3.  6a^-ic-15.  15.    6w2-5w-6. 

4.  6m2 -5m -25.  16.   9  a2- 9a- 10. 

5.  Up2  -  39  p  -  35.  17.   12  a2  -  31  a  -  15. 

6.  10v2-29v-21.  18.   36?/2 -f-7^- 15a;2. 

7.  2a2-4a  +  2.  19.   8z2-38x  +  35. 

8.  m2  +  7£-12.  20.    (m  +  7i)2-ll(??i+w)_26. 

9.  6s2-s-12.  21.    15  #2  + 29  a -14. 

10.  2b2-6b-8.  22.    12w2-31ti-15. 

11.  4  +  ±y-15y2.  23.    (p2+p)2-  U(p2+p)+2±. 

12.  2z2-5a;-3.  24.    6a2-a6-3562. 

25.  (a  +  6)2-3(a  +  6)-54. 

26.  5aj2  +  io^  +  52/2+20(a;  +  2/)-105. 

27.  a2  -  (p  -f  l)<?a2  +  M2a2. 

28.  10a2»  +  31a»  +  15. 


150  Factoring 

REVIEW  EXERCISE 

229.  Factor  the  following  : 

1.  a-l  +  6(a-l).  12.  2  m2  -  20  m  +  50. 

2.  m2(l-n)-p2(n-l).  13.  6-9n+3n\ 

3.  16a%2-20a,7/.  14.  5  a2  +  30  a;  +  40. 

4.  3p  +  3  q  -  ap  -  aq.  15.  2(q  +  l)2  -  8. 

5.  10*3-7*2-12*.  16.  l+y-2x-2xy. 

6.  8 a  -  14 aa?  —  15 a#2.  17.  2xy  —  2x-\-y— 1. 

7.  6x2y  +  ±6xy-16y.  18.  2m  -  3n  +  2m2  -3mn. 

8.  16ic2-9i/2.  19.  2a;2+4x?/+22/24-2a:+2y/. 

9.  2  a;2 -8.  20.  z2  +  fa?-f. 

10.  2  a;2 +  6  a; +  4.  21.    (ax  -  by)  -  (ax  -  by)*. 

11.  3 -27  s2.  22.    (a;2  -  5  a;)2 -(a;2-5  a;) -20. 

23.  a2-62-4a2  +  8a&-462. 

24.  am  +  bn  +  cp  -f  6m  +  en  -f  ap  +  cm  +  an  -f-  6p. 

25.  7(a2  -  b2)  -  7 (a  +  b)2  +  7 (a  +  6). 

26.  (a2  -  b2)(a  +  b)-  2(a2  -  b2)  +  (a  -  6). 

27.  6(s2  -  £2)(s  -  0  +  42(s2  -t2)+72s+  72  fc 

28.  r(p  —  q)2  —  4  r(p  —  a)  —  12  r. 

230.  Case  VIII.  Type  Forms  a3  -f  6s  and  a3  -  ft3,  —  Sum  or 
Difference  of  the  Cubes  of  Two  Numbers. 

By  actual  division  we  obtain  the  following : 

1.  (a3  +  63)-i-(a+&)=a2-a&+  b2. 

2.  (a3-b3)  +  (a-b)=a2  +  ab  +  b2. 

From  1  we  may  state  the  following : 

The  sum  of  the  cubes  of  two  numbers  is  always  divisible  by  the  sum 
of  the  numbers.  The  quotient  is  the  square  of  the  first  number  minus 
the  product  of  the  two  numbers  plus  the  square  of  the  second  number. 

Let  the  student  make  the  corresponding  statement  for  2. 


Factoring  151 

ORAL  EXERCISE 

231.  Divide  the  following  as  indicated  in  §  230: 

i.  (s?-ft+(p-q)- 

2.  (rf+y^  +  ^  +  y). 

3.  (8m3-27w3)--(2  7/i-3w). 

4.  (64a3  +  12563)_j-(4a+5&). 

5.  (27s3  -*3) -s-(3  8-0- 

6.  (8p3^3+?'3)--(2jp(?  +  r). 

7.  Q^  +  ^-^  +  y). 

8.  (125d3  +  64ft3)-K5d4-4fc). 

9.  (l-27o6)--(l-3a;2). 

10.  (m6  +  n«) +(«£  +  »*). 

11.  (125 -a6)-(5- a2). 

12.  (^-y9)^-(x2-f). 

232.  From  §  230  we  may  make  rules  for  factoring  the  sum 
or  the  difference  of  the  cubes  of  two  numbers.  These  rules 
in  algebraic  symbols  are : 

1.  a3  +  63=(a  +  &)(a2-a&  +  &2). 

2.  a*-V=(a-b)(d*+ab  +  V*). 

In  words  we  have  from  1 : 

The  sum  of  the  cubes  of  two  numbers  equals  the  sum  of  the  numbers 
multiplied  by  the  square  of  the  first  number  minus  the  product  of  the  two 
numbers  plus  the  square  of  the  second  number. 

Let  the  student  make  the  corresponding  statement  for  the 
difference  of  the  cubes  of  two  numbers. 

Example 
Factor  afy8  —  27. 

Solution.  xsys  —  27  may  be  written  (xy)&— 33.  It  therefore  comes 
under  (2)  and  we  have 

(xyy  -  33  =  (xy  -  3)  (aty  +  3xy  +  9). 
In  this  example,  what  replaces  a  of  the  type  form  ?     What  replaces  b  ? 


8. 

64&3a?  +  27c32/3. 

9. 

v3- 

w3. 

10. 

i>3- 

i 

12  5"' 

11. 

**- 

- 125  in3n3. 

12. 

32  aJ 

!-10863. 

13. 

Z2ra3 

-  jy. 

14. 

Z>6- 

64. 

(m2)3  - 

(n2)3.      Factor 

both 

152  Factoring 

EXERCISE 

233.  Factor  the  following : 

1.  a363  +  8. 

2.  a3 +  27. 

3.  ra3-l. 

4.  m6  +  ^=(ra2)3+(<c2)3. 

5.  125  -a3b3. 

6.  1+v3. 

7.  a3&3  —  m3n3. 

15.  m6  -  n6  =  O3)2  -  (n3)2    or    (m2)3  -  (w2)3. 
ways.     Which  way  is  to  be  preferred  ? 

16.  l  +  6x  +  6a?  +  a?=(l  +a3)-f  6oj(l  +  a;) 

=(1  + «)  [(1  -  x  +  a;2)  +  6a;]  etc. 

17.  a3  +  5a2  +  5a  +  l.  24.  a4  -  a (x  +  y)3. 

18.  a3  +  3a2a;  +  3aa;2  +  a53.  25.  (a;  +  y)4  - (x  +  y). 

19.  (a-h&^  +  c3.  26.  m9-a6. 

20.  (a-6)3-c3.  27.  64a^-l. 

21.  a3-(b-c)3.  28.  a3  -  3  a26  +  3  ab2  -  ft3. 

22.  (a;2  +  2/2)3  +  8.  29.  a3n  -  63n. 

23.  (1  —  2a)3+'l.  30.  a3n  +  b3n. 

SUMMARY   OF   FACTORING 

234.  The  following  summary  of  the  type  forms  in  factoring 
is  inserted  here  for  convenience  and  review : 

I.    Factors  of  monomials.     Square  root. 

II.   Terms  having  a  common  monomial  factor. 

ab  +  ac  =  a(b  +  c). 


III.   The  square  of  a  binomial. 


a2  4-  2  ab  +  &  =  (a  +  &)2, 
a2-  2  ab+&  =  (a  -b)\ 


Summary  of  Factoring  153 

IV.   The  difference  of  the  squares  of  two  numbers. 

a*-&=(a  +  b)(a-b). 

IV  a.   Polynomials  written  as  the  difference  of  two  squares. 

(a2  +  2  aft  +  ft2)  -  c2  =  (a  +  ft  +  c)(a  +  &  -  c). 

IV  ft.   Completing  the  square. 

**+ *2**+y*=x4+2  x8^+y*-jre|^=(jca+jry+»«)(jra-*yH-»B). 

V.   Grouping  terms.     a(*-f  #)  +  &(* +y)  =  (jr  4- #)(<*+ ft). 

VI.   The  quadratic  trinomial  of  the  form  x2+bx+c. 

VII.   The  general  quadratic  trinomial,     ax2  +  bx  +  c. 

VIII.   The  sum  or  the  difference  a3  +  ft3  =  (a  +  &)(a2  -  a&  +  ft2) 
of  two  cubes.  a3  -  ft3  =  (a  -  &)(a2  +  ab  +  ft2). 

Let  the  strident  translate  each  algebraic  formula  into  ver- 
bal language. 

235.  The  student  finds  it  comparatively  easy  to  factor  al- 
gebraic expressions  when  they  are  classified  under  proper  type 
forms,  but  in  actual  practice  he  is  left  to  his  own  devices  to 
determine  to  what  type  form  a  given  expression  belongs. 
Hence  he  usually  meets  with  difficulty  in  factoring  unclassified 
expressions. 

In  factoring  miscellaneous  exercises  the  student  will  find 
the  following  suggestions  useful : 

1.  Carefully  study  the  type  forms  and  determine  which  one  applies  to 
the  given  expression. 

2.  If  there  is  a  monomial  factor,  write  the  expression  as  the  product 
of  the  monomial  factor  and  a  polynomial.     (II.) 

3.  If  the  polynomial  is  a  binomial,  note  whether  it  is  the  difference  of 
two  squares  or  the  sum  or  the  difference  of  two  cubes.     (IV,  VIII.) 

4.  If  the  polynomial  is  a  trinomial,  use  the  proper  type  form  for  fac- 
toring a  trinomial.     (Ill,  IV  b,  VI,  VII.) 

5.  If  there  are  more*  than  three  terms,  first  try  to  factor  by  grouping 
the  terms.  It  is  also  possible  that  the  terms  can  be  arranged  as  the 
difference  of  two  squares.     (IV,  IV  a.) 


i54  Factoring 

6.  Continue  the  process  of  factoring  till  all  the  factors  are  prime. 

7.  Check,  either  by  multiplying  the  factors  together,  or  by  substitut- 
ing definite  values  for  the  letters  both  in  the  indicated  product  of  the 
factors  and  in  the  original  expression.  In  the  latter  case  the  two  lesults 
should  be  equal. 

REVIEW  EXERCISE 

236.   Factor  into  prime  factors  : 

1.  a362  +  62.     (3  factors.)  9.  a6&2-aW-a4a4+6a;7. 

2.  a4  -  2  x2  +  1.     (4  factors.)  10.  a2  +  21  a  +  108. 

3.  54  tftfz9  -  42  a5*5  -  24  xy7z7.  11.  729-a%12.    (4  factors.) 

4.  512  afy3-  27*9.  12.  62+12  6  +  35. 

5.  aV-2a%  +  66.  13.  a4+arJ  +  «  +  l- 

6.  1  -  14  x*y  +  49  x*y2.  14.  a2  +  2  a?>  -  15  b2. 

7.  2  a;7?/ -3  ay +  5  ay.  15.  81  a2  -  16(2  a  -  3  xf. 

8.  z4  +  22z2+169.  16.  (2  a-  36)2  -  4  b\ 

17.  21  a869  -  28  ab¥  +  35  a367c. 

18.  (a  +  &)2+3(a  +  6)-4. 

19.  asc  —  bx  +  ca  +  cm/  —  by  +  cy. 

20.  2ax  —  5ay  +  a  —  2bx  +  5by  —  b. 

21.  (4a-5  6)(5c-2d)-(a  +  46)(5c-2d). 

22.  asc-a+x-1.  33.  8 a4 -or2 -9. 

23.  a2  -  62  +  2  6c  -  c2.  34.  12  a2  -  17  a&  +  6  62. 

24.  a2  —  (c  +  5)  a  +  5  c.  35.  a62c  +  bcx  +  a&i/  +  xy. 

25.  a2  +  (a-6)a;-a6.  36.  -  x2  +  2  a  -  1. 

26.  x2-(n-3)x-3n.  37.  a4  +  4  64. 

27.  64  +  27  63.  38.  a6-66. 

28.  125  x*-8f.  39.  a8-68. 

29.  #y  -81m4.  40.  a12  -  612. 

30.  a4  +  2  ajy  -  15  2/4.  41.  8a4-^2-9^. 

31.  3^  +  8^  +  5.  42.  12x2  +  2xy-30y2. 

32.  2  a2  + 13  a  + 15.  43.  m4  +  4  m2p2  +  100  p4. 


Review  Exercise  155 

44.    2r2+16rs  +  32s2.  45.    (a  +  b)2  -  a2  +  b2. 

46.  (x  —  y)2  —  xrz-\-  y2z. 

47.  5  tfy  -  23  x2y2  +  12  xy\ 

48.  a62c  -|-  2  fcesc  —  afo/  —  2  a;?/. 

49.  (a  +  &)2+(a2-62,+  a  +  &. 

50.  64  x2  +  81  y2-  144  an/. 

51.  18  a4  +  72  wi4.     (3  factors.) 

52.  8  a3 +  64  a?.  55.    125a*»-y». 

53.  4a2-12a&  +  962-9.  56.   4  -  12  ab  -  4  a2  -  9  b\ 

54.  18  a262+32  a*+48aWV  57.    a  -  a7. 

58.  (a  +  6)3  -  (a2  -  b2)  (a  +  6)  +  a  (a  +  6)2. 

59.  (z  +  2/)4-(x  +  2/)*. 

60.  36  a462c2  -  24  abbc2  +  72  a363c. 

61.  30  arty3  -  45  x y  +  60  an/5. 

62.  (a-b)2-(a-b). 

63.  z2-z-156. 

64.  12  a&z2  +  48  a2&2an/  -f  48  a3&y. 

65.  6  a*  -  10  a4a;  -  18  a2a;2  + 30  ar>. 

66.  3  a-3 +  3  a-2 -36  a;. 

67.  9  a-2  -  4  y2  +  4  yz  -  z2. 

68.  (a  +  3  6)2-9(6-c)2. 

69.  a2(l-c)-4&2(c_i). 

70.  a?/ (a;  —  m)—ax (y  —  m).     (First  expand.) 

71.  ay{x  —  m)—ax(m  —  x). 

72.  a2a;  -f-  ab2x  —  afo/  —  Wy. 

73.  3  (m  +  n)3  -  4  (m  +  n)2  +  ra  +  w. 

74.  4(a  -  6)3  +  12  a(a  -  6)2  -  6(a  -  6)a2. 

75.  2  w(2  m  -  1)  -  3  rc2(2  m  -  1)  +  5(2  m  -  1). 

76.  2  ax  -h  3  6a;  -f-  4  ca;  —  2  a#  —  3  by  —  4  cy. 

77.  2  a#  —  3  bx  +  4  ca;  +  2  ay  —  3  by  +  4  c#, 


1 56  Factoring 

Factor  : 

78.  2  ax  —  3  bx  +  4  ex  —  2  ay  +  3  &#  —  4  cy. 

79.  27aaj3  +  8ay. 

80.  aW  -bz-a2  +  l. 

81.  (a  —  b)(x  —  y)  —  (a  —  y)(x  —  b).     Expand. 

82.  (a  +  6)2-  4  -  2(a  +  6  -  2). 

83.  a2b  +  62c  +  c2a  -  ab2  -  be2  -  ca2. 

84.  Factor  x6  —  2/6  into  two  factors  in  two  different  ways 

85.  a6  +  &6.  93.    a4n-l. 

86.  a10-b10.     (2  factors.) 

87.  a12  -b12. 

88.  a2n-l. 

89.  a2n-b2n. 

90.  a2n-a2. 

91.  a2n+1  —  a. 

92.  a2n  +  2a"H-l. 

THE  SOLUTION  OF  EQUATIONS  BY  FACTORING 

237.  A  root  of  an  equation  has  been  defined  as  a  value  of 
the  unknown  quantity  that  satisfies  the  equation. 

Is  2  a  root  of  x2  —  5  x  +  6  =  0  ?  is  la  root  ?  is  6  a  root  ? 
is  3  a  root  ? 

238.  It  is  sometimes  possible  to  write  an  equation  in  such  a 
form  that  its  roots  are  evident. 

1.    Consider  the  equation,  x2  —  7  x  -f- 10  =  0. 

Factoring  the  first  member,  we  have  the  same  equation  in  another  form. 

(a:_5)(x_2)=0. 
If  the  product  of  two  or  more  factors  is  zero,  one  of  the  factors  must 
be  zero.     Therefore  this  equation  is  satisfied,  if 

x  —  5  =  0 ;  that  is,  if  x  —  5, 
or  if  x  —  2  =  0  ;  that  is,  if  x  —  2. 

Therefore,  5  and  2  are  roots  of  this  equation.     The  roots  may  be  verified 
by  putting  5  and  2  for  x  in  the  equation. 


94. 

an+1  +  2anb  +  a  +  2b. 

95. 

(a  —  b)m+1—(a  —  b)m. 

96. 

#3«+l  _  x. 

97. 

x2n  —  y2n  —  xn  —  yn. 

98. 

x2n  —  y2n  -f-  xn  +  y*. 

99. 

a18  -  1. 

100. 

a32  -  1. 

The  Solution  of  Equations  by  Factoring      157 

2.    Solve  z3  =  4  x. 

Make  the  second  member  zero  by  transposing  4  x. 

x3-4z  =  0. 
x(x  —  2)(x  +  2)  =  0.     (Factoring.) 
The  second  equation  is  satisfied  if  any  one  of  the  three  linear  equations, 
x=0,  z-2  =  0,  z  +  2  =  0, 
is  satisfied.     (Why  ?) 

This  gives  as  solutions  of  the  given  equation  x  =  0,  2,  or  —  2. 
Verify  by  substituting  0,  2,  and  —  2  for  x  in  the  original  equation, 
x?=4x. 

We  can  solve  equations  by  factoring  if,  when  the  second 
member  is  zero,  we  can  factor  the  first  member  into  factors  of 
the  first  degree  with  respect  to  the  unknown  number. 

239.    To  solve  an  equation  by  factoring : 

1.  Write  the  equation  with  the  second  member  zero  and  the  first 
member  arranged  in  descending  powers  of  the  unknown  number. 

2.  Factor  the  first  member  into  linear  factors  with  respect  to  the 
unknown  number. 

3.  Put  each  factor  equal  to  zero  and  solve  the  linear  equations  ob- 
tained. 

Examples 

1.    Solve2«(a;-l)=3a-2. 


Solution.            2x2  —  2x  =  Sx- 

2.     (Multiplying.) 

2z2_5a;_f2  =  0. 

(Transposing.) 

(2E-l)(Z-2)=:0. 

(Factoring.) 

2z-l  =  0 

or  x  =  \. 

x-2  =  0 

or  x  =  2. 

Let  the  student  verify  the  roots. 

2.    Solve9(z-l)  =  (a;  +  4)(a: 

-i). 

Solution. 

9(x-l)-(x  +  4)(x-l)=0. 

(Transposing,  without  multiplying.) 

(x-  l)(9-s-4)=0. 

(Factoring.) 

(z_l)(5-a-)  =  0. 

x  —  1  =  0  or  ce  =  l. 

5  —  £=0or:r:=5. 

Verify  by  substituting  the  roots  in  the  equation. 

158  Factoring 

EXERCISE 

240.    Solve  the  following  equations  : 

1.  (a; -5)0- 4)=  0.  7.    x2-6x  =  T. 

2.  O  +  i)(7a;-l)=0.  8.   a;2  =  13a;-42. 

3.  (2x-5)(x-3)=0.  9.    (a; -1)(« -2)=  12. 

4.  o?(3a;-7)=0.  10.    9  x2  -  16  =  0. 

5.  (4-aj)(5aj  +  l)=0.  11.   4a;2  -4a  +  1  =  0. 

6.  z2-7a=-10.  12.    ?y(?y_6)=72/-42. 

13.  (2, -11)0/ -12)=  2. 

14.  (r+6)(r-4)-(2  +  r)(2-r)=56. 

15.  0- l)2 +(^+ 1)2  =  29 -(2a: -f-3)2. 

16.  2a;2- 5a;  =  3. 

17.  (x  -  2)2-  (x  +  2)2  +  7 x  =  0. 

18.  (3a:-5)(3a:  +  5)-(a;-l)2  =  10.  ♦ 

19.  0+  l)3-3a;(a;-l)=a^-|-l. 

20.  (a>  +  2)3-2(a;+2)2  =  0. 

21.  x2  —  ax  —  bx  +  a&  =  0. 

22.  a;2  —  4  a2  -  4  a  —  1  =  0. 

23.  a;3  +  a;2=a;  +  l. 

24.  O-2)2+25  =  10O-2). 

25.  (x-7)(2a;  +  5)=(3a;-l)(a;^7). 

26.  (a;-3)(4a;-5)  =  a:2-9. 

27.  x~  —  9=  8x. 

28.  (2  as  —  f)  -4=(2a;-f)(5a;-ll). 

29.  0_2)  =  0-2)(a;-3). 

30.  (aj  -  1)0  -  2)(a;  -  3)  +  6  =  0.     (Find  one  root  only.) 


The  Solution  of  Equations  by  Factoring      159 

SOLUTION   OF  PROBLEMS 
241.    1.    The  larger  of  two  numbers  exceeds  the  smaller  by 
5,  and  their  product  is  84.     Find  the  numbers. 

Solution.  .  Let  x  =  the  smaller  number. 

Hence  x  +  5  =  the  larger  number, 
and  x(x  +  5)  =  their  product. 
Then  x(x  +  5)  =  84.     (By  the  conditions.) 
Hence  x2  +  5  x  -  84  =  0,     (Why  ?) 
or  (a;  +  12) (a;  -  7)  =  0.     (Why  ?) 

.-.  x  +  12  =  0  and  x  -  7  =  0. 

.-.  x--  12  or  7, 
and  x  +  5  =  -  7  or  12. 
The  pairs  of  numbers  that  satisfy  the  conditions  of  the  problems  are 
—  12  for  the  smaller  and  —  7  for  the  larger,  or  7  for  the  smaller  and  12 
for  the  larger. 

To  check  the  answers  they  should  be  put  into  the  original  problem  and 
not  into  the  equation.     (Why  ?) 

2.  The  length  of  a  rectangular  figure  is  5  inches  more  than 
its  width,  and  its  area  is  84  square  inches.  Find  its  dimensions. 

Solution.  Let  x  =  the  number  of  inches  wide. 

Hence  x  +  5  =='  the  number  of  inches  long, 
and  x(x  +  5)  =  the  number  of  square  inches  in  area. 
Thenx(a-  +  5)  =  84. 
From  this  point  the  solution  is  exactly  like  that  of  the  last  problem. 
The  answers  —  1 2  and  7  as  values  of  x  have  to  be  considered  in  connec- 
tion with  the  problem.     The  answer  —  12  cannot  represent  the  number 
of  inches  in  the  width  of  a  rectangle  and  is  to  be  rejected  in  this  problem. 
x  =  7  is  evidently  the  answer  to  be  used.     This  will  make  the  dimensions 
of  the  rectangle  7  inches  and  12  inches. 

3.  The  product  of   two  consecutive  numbers  is  72.     Make 
the  equation  and  solve  for  the  numbers. 

4.  The   product   of  two   consecutive   even  numbers  is  80. 
Make  and  solve  the  equation  to  find  the  numbers. 

5.  The  sum  of  two  numbers  is  19  and  their  product  is  84. 
Find  the  numbers. 

Hint.     Let  the  numbers  be  represented  by  x  and  19  —  x. 


160  Factoring 

6.  One  of  two  numbers  is  twice  as  large  as  the  other  and 
their  sum  is  14.     Find  the  numbers. 

Hint.  Let  x  and  2  x  represent  the  numbers.  The  equation  is  of  first 
degree. 

7.  One  of  two  numbers  is  twice  the  other  and  their  product 
is  242.     Find  the  numbers. 

8.  A  rug  is  twice  as  long  as  it  is  wide.     It  contains  4^- 
square  yards  of  material.     Find  its  dimensions. 

9.  The  perimeter  of  a  rectangle  is  40  inches  and  its  area  is 
91  square  inches.     Find  the  dimensions. 

Hint.  If  the  perimeter  is  40  inches,  the  sum  of  the  length  and  the  width 
is  20  inches. 

10.  The  side  of  one  square  is  4  inches  more  than  that  of 
another  and  the  sum  of  their  areas  is  136  square  inches.  Find 
the  side  of  each  square. 

11.  If  this  page  is  6  centimeters  longer  than  it  is  wide,  and 
its  area  is  216  square  centimeters,  find  the  dimensions. 

12.  The  quotient  exceeds  the  divisor  by  8  and  the  dividend 
equals  three  times  the  sum  of  the  divisor  and  the  quotient. 
Find  the  divisor,  the  quotient,  and  the  dividend. 

13.  The  sum  of  the  squares  of  two  consecutive  numbers 
exceeds  5  times  the  sum  of  the  numbers  by  6.  Find  the 
numbers. 

14.  A  rectangle  is  3  inches  longer  than  it  is  wide.  If  both 
dimensions  are  increased  by  2  inches,  the  area  is  28  square 
inches.     Find  the  original  dimensions. 


IX,   HIGHEST  COMMON  FACTOR  AND  LOWEST 
COMMON  MULTIPLE 

HIGHEST  COMMON  FACTOR 

242.  Rational  Term.  A  term  is  rational  if,  when  reduced  to 
its  simplest  form,  it  contains  no  indicated  roots. 

Thus,  Sac2,  3a26,  and  Viare  rational  terms.  7  ay/b  is  not  rational 
with  respect  to  ft  but  it  is  rational  with  respect  to  7  and  a.    Is  V27  rational  ? 

^27? 

243.  Integral  Term.  A  term  is  integral  with  respect  to  any 
set  of  numbers  or  letters  if  none  of  the  numbers  or  letters 
appear  in  the  denominator. 

Thus,  lab  is  integral  with  respect  to  7,  a,  and  b  but  not  with  respect 
to  3.      4mnspq2   is  integral  with    respect  to    all   letters    and    numbers 

involved.     — —  is  not  integral  with  respect  to  c. 

c  ... 

244.  A  polynomial  is  rational  and  integral  if  all  its  terms 
are  rational  and  integral. 

Thus,  3  ax  +  2  by  is  rational  and  integral. 

2  ft 
3  ax  H is  rational  but  is  not  integral. 

y 

3  ax  +  V2  by  is  integral  but  is  not  rational. 

An  expression  may  be  rational  and  integral  with  respect  to 

some  particular  letter  involved.     The  three  examples  just  given 

are  all  rational  and  integral  with  respect  to  x. 

b  c 

Also  x2  +-  -  x  +  -  is  rational  and  is  integral  with  respect 
a  a 

to  x}  b,  and  c,  but  is  not  integral  with  respect  to  a. 

161 


162  Highest  Common  Factor 

245.  Degree.     The  degree  of  a  rational  integral  monomial  is 

the  sum  of  the  exponents  of  its  literal  factors. 

Thus,  2  ax  and  a3  are  of  third  degree  ;  4  ab2c  is  of  fourth  degree.  Of 
what  degree  is  7  x^  ?     Sax?    2  a2x2  ? 

We  are  sometimes  concerned  with  the  degree  of  a  monomial 
with  respect  to  some  particular  letter. 

Thus,  3  a'2x  is  of  the  second  degree  with  respect  to  a.  It  is  of  the  first 
degree  with  respect  to  x. 

246.  The  degree  of  a  rational  integral  polynomial  is  the  same 
as  that  of  its  term  of  highest  degree. 

Thus,  3  xs  +  2  x2y2  is  of  the  fourth  degree.  Of  what  degree  is  ax2  + 
bx+c?  of  what  degree  with  respect  to  x?     with  respect  to  a  ?     b?    c? 

247.  The  student  should  note  that  degree  and  power  are  not 
the  same.  The  power  of  a  term  may  or  may  not  be  the  same  as 
the  degree  of  the  term. 

Thus,  3  x2y2  and  a4  are  both  of  the  fourth  degree,  but  3  x2y2  is  not  a 
fourth  power.  Also  (a2  +  2)2  is  a  second  power,  but  is  a  fourth  degree 
expression. 

ORAL  EXERCISE 

248.  Give  the  degree  of  each  of  the  following  : 

1.  3  a2by.  '  5.  ax2  +  bx  +  c. 

2.  4  obex.  6.  axy  +  by3. 

3.  2  xhf.  7.  (x  +  y)2. 

4.  5  axy.  8.  (x  +  y)\ 

9.    (x  +  l)3  +  a(x  +  l)2  +  b(x  +  1). 

Of  what  degree  is  each  of  the  above  with  respect  to  a;? 
with  respect  to  x  and  y  ? 

249.  Common  Factor.  If  the  same  factor  occurs  in  two  or 
more  algebraic  expressions,  it  is  a  common  factor  of  the 
expressions. 

Thus,  x  is  a  common  factor  of  7  a;  and  3  xy  ;  and  2  a  is  a  common  factor 
of  2  a,  4  a,  and  6  a8. 


Highest  Common  Factor  163 

250.  Two  or  more  expressions  may  have  several  common 
factors. 

Thus,  35  x3y2,  21  x2yB  and  42  xsys  have  what  common  factors  of  the 
first  degree  ?  of  the  second  degree  ?  of  the  third  degree  ?  of  the  fourth 
degree  ?  Can  you  find  a  common  factor  of  these  expressions  of  higher 
degree  than  the  fourth  ? 

251.  The  highest  common  factor  (H.  C.  F.)  of  two  or  more 
monomials  is  the  greatest  common  divisor  of  their  numerical 
coefficients  multiplied  by  their  highest  degree  literal  common 
factor. 

Thus,  7  x2y2  is  the  H.  C.  F.  of  35  x*y2,  21  x2y*,  and  42  xsy*. 

252.  The  H.  C.  F.  in  algebra  corresponds  to  the  greatest 
common  divisor  (G.  C.  D.)  in  arithmetic.  The  G.  C.  D.  is  the 
largest  number  that  will  exactly  divide  two  or  more  numbers ; 
the  H.  C.  F.  is  the  highest  degree  algebraic  expression  that  will 
divide  two  or  more  expressions. 

We  may  find  the  G.  C.  D.  of  12,  18,  24  by  factoring  thus  : 
12  =  22  •  3,  18  =  2  •  3*,  24  =  23  .  3. 

Therefore  the  G.C.D.  of  12,  18,  and  24  is  2  .  3  =  6. 

Similarly,  we  may  find  the  H.  C.  F.  of  two  or  more  algebraic 
expressions. 

Find  the  H.  C.  F.  of  12  a?bc,  18  aWc2,  24  a?c. 
Solution.  12  a2bc  =  22  •  3  •  a2bc. 

18a3W  =  2.32-aW. 
24  a?c  =  23  •  3  •  asc. 
The  H.  C.  F.  is  the  G.  C.  D.  of  the  numerical  coefficients,  6,  multiplied 
by  their  highest  degree  literal  common  factor  a?c ;  that  is,  the  H.  C.  F.  is 
6  a2c. 

253-  To  find  the  H.  C.  F.  of  two  or  more  algebraic  expressions,  multiply 
together  the  lowest  powers  of  all  the  prime  factors  common  to  all  the 
expressions. 

In  the  case  of  monomials  the  H.  C.  F.  is  seen  by  inspection. 
If  any  of  the  expressions  are  polynomials,  factor  them  into 
prime  factors. 


164  Highest  Common  Factor 

EXERCISE 

254.     Find  the  H.  G.  F.  in  each  of  the  following : 

1.  3a2&,  6aW,  9ab2. 

2.  4a¥,  6aW,  12  ax2. 

3.  Ua2bW,  98ff362a4,  105  a'W5. 

4.  45  a6,  18a567,  108  a4612. 

5.  13  x*y\  52  a3?/6,  169  x*y\ 

6.  3a%3,  9  xy,  12  a?y,  15  a%5. 

7.  4a%2,  16  #y,  64  a%4. 

8.  98  a¥,  180  arV,  300  sc4*5. 

9.  15  a*bxy,  45  6s?/4,  90  a^x4. 

10.  14(a  +  b)\a  -  b),  10(a  +  &)< 

11.  a26  -  63,  a2b  -  2  ab2  +  b3,  a*b  -  ab*. 

Solution.  a?b  —  68  =  6(a  +  &)(a  —  b). 

a2b-2ab2  +  b*  =  b(a-b)2. 

a*b  -  a6*  =  ab(a  -  b)(a2  +  ab  +  62). 
.-.  H.C.F.  =  6(o-6;. 

.      12.  24  a2x2  -f  36  a3x*,  9  a£  -  12  a2x2. 

13.  3  aa?  +  4  to4,  ax5  - 12  to6. 

14.  4a262a;2-8ato3,  8  a2bx*  -  12  abx2. 

15.  18  a26V  -  72  a8,  12  ato4. 

16.  2x2-17a+36,4a2-12a-27. 

17.  (a  +  &)2-  c2,  a2-(6  +  c)2. 

18.  9  a?4-  16  y4,  9  a2a2  +  12  a2?/2. 

19.  4a2+12a'?/+92/2,  16a  +  24#. 

20.  (a  +  b)3,  a2  +  2  a&  +  &2,  «2  -  &2- 

21.  48  a4  -  12  s/4,  20  x3  -  10  a;?/2. 

22.  a4  -  &4,  a3  -  53,  a2  -  ?>2. 

23.  a?2-5a  +  6,  3a;2- 6a?,  a;2  -  6  a;  +  8. 

24.  3a;2-a;-2,  6a;2  +  13a;  +  6,  6a;2-5a5- 6. 


Lowest  Common  Multiple  165 

25.  2  a2b  +  2  a&2  -  2  ad^  3  6c2  -  3  62c  -  3  abc. 

26.  a2  +  b2  -  c2  +  2  a&,  a2  -  62  +  c2  +  2  ac. 

27.  a2  —  62  —  ac  +  6c,  a&  +  ac  +  62  —  c2. 

28.  mx  —  m  —  x  +  1,  m2  —  2  m  +  1. 

29.  2a6-  3ac- 26  +  3c,  3a6-2ac-36+2c. 

30.  x2  -  x  -  20,  x2  +  a  -  30,  x*  -  25. 

LOWEST  COMMON   MULTIPLE 

255.  A  product  is  a  multiple  of  any  of  its  factors. 
Thus,  3  x2y  is  a  multiple  of  as ;  of  xy  ;  of  3  x  ;  etc. 

256.  A  common  multiple  of  two  or  more  expressions  is  a 
multiple  of  each  of  them. 

Thus,  6  x2y3  is  a  common  multiple  of  3  x,  2  y  and  xy. 

Two  or  more  expressions  have  always  an  infinite  number  of 
common  multiples. 

Thus,  3x,  2y,  and  xy  have  as  common  multiples  6xy,  6x2y2,  6x2y, 
12  xy,  etc.,  indefinitely.  Can  you  find  a  common  multiple  of  these  three 
monomials  of  lower  degree  than  the  second  ? 

257.  The  lowest  common  multiple  (L.  C.  M.)  of  two  or  more 
monomials  is  the  arithmetical  least  common  multiple  of  their 
numerical  coefficients  multiplied  by  their  lowest  degree  literal 
common  multiple. 

258.  In  arithmetic  the  least  common  multiple  of  two  or  more 
numbers  is  the  smallest  number  which  may  be  exactly  divided 
by  each  of  them.  In  algebra  the  L.  C.  M.  of  two  or  more  ex- 
pressions is  the  lowest  degree  expression  which  may  be  ex- 
actly divided  by  each  of  them. 

259.  To  find  the  L.  C.  M.  of  two  or  more  algebraic  expressions,  mul- 
tiply together  the  highest  powers  of  all  the  different  prime  factors  in 
the  expressions. 


166  Lowest  Common  Multiple 

The  L.  C.  M.  of  monomials  is  seen  by  inspection.     If  the 
expressions  are  polynomials,  first  factor  them  into  prime  factors. 

1.  Find  the  L.  C.  M.  of  9  b\  12  ac2,  4  abc\ 

Solution.     9  b8c  =  32b%  12  ac2  =  3  .  2'2ac2,  4  abcs  =  2%&c3. 
.-.  L.  C.  M.  =  32  •  2Wc3  or  36  ab8c*. 

2.  Find  the  L.  C.  M.  of  a*-3a  +  2,  a2  -  1,  a2-4a  +  4. 

Solution.  a2  -  3  a  +  2  =  (a  -  l)(a  —  2). 

^1=  (a+l.)(a-l). 
a2- 4a +  4  =  (a  -  2)2. 
.-.  L.C.M.  is(a-l)(a  +  l)(a-2)«. 

EXERCISE 

260.    i^md*  £/<e  Z/.  C.  M.  in  each  of  the  following,  leaving  the 
results,  in  the  case  of  polynomials,  in  factored  form : 

1.  3  ab,  4  a2bc,  6  ab2c.  6.  m  -+-  n,  (m  —rif,  m2—n2. 

2.  12,  18,  24  x.  7.  a2 -6  a&  +  9  b2,  a2-9  62. 

3.  2  a,  3  6,  5  c.  8.  3  a+6,  6a;2-24,  2a-4. 

4.  2  a,  3  a,  5  a.  9.  3  —  3  a2,  5  —  5  a,  1  -f  a. 

5.  x*  + 1,  2  *  -  2,  a2  - 1.  10.  z3  - 1,  (a?  -  l)2. 

11.  a3  -  a,  a2  -  2  a  +  1,  2  a2  -  5  a  +  3. 

12.  2  a2  -  5  a  +  3,  4  a2  -  13  a  +  3,  8  a2  -  6  a  + 1. 

13.   What  is  the  L.  C.  M.  of  two  expressions  that  have  no 
common  factor  ? 

14.  2x2-\-  x-  1,  x2-x  -2,  2x2-5x+2. 

15.  2(2  a; +  5),  3x  +  6,  2  a2 +  9  a; +10. 

16.  a2  +  3  x  +  2,  x2  +  4  a;  4-  3,  a2  +  5  a?  +  6. 

17.  a2  -  3  ab  +  9  62,  a3  +  27  63,  a  +  3  b. 

18.  6  -  a  -  a2,  2  -  3  a  +  a2,  1  -  a. 

19.  xy-2y2,xy-y2,x*-3xy  +  2y2. 

20.  2  -  2  a;  —  a  -f  a#,  3  —  3  x  -  b  -f  bx. 

21.  3  a2  —  5  ax  +  2  .x2,  4  a2  —  9  aa  +  5  x2. 


X,  FRACTIONS 
261.   An  algebraic  fraction  is  an  indicated  division. 

Thus,  ^  (read  the  fraction,  a  divided  by  b)  is  the  indicated  quotient  of 
a  divided  by  b. 

The  numerator  of  the  fraction  is  the  dividend  and  the  de- 
nominator is  the  divisor. 

Terms  of  a  Fraction.  The  numerator  and  the  denominator 
are  the  terms  of  a  fraction.  The  denominator  of  a  fraction  can- 
not be  0  since  dividing  by  0  has  no  meaning  in  the  ordinary 
sense  of  division. 

The  topics  studied  under  fractions  in  algebra  agree  closely 
with  those  of  arithmetic,  and  the  methods  are  similar. 

EXERCISE 

262.    1.    Reduce  J-f  to  lowest  terms.     Also  |-§. 

2.  Change  the  improper  fraction  *£■  to  a  mixed  number. 
Give  the  rule. 

3.  Change  f ,  f ,  -J  to  equivalent  fractions  having  the  least 
common  denominator.     Give  the  rule. 

4.  State  the  rule  for  adding  arithmetical  fractions. 

5.  lf  +  3i  +  3J  =  ? 

6.  75£-12f  =  ? 

7.  Multiply  21  --  f  by  |  of  |  x  f . 
Note.     For  order  of  operations  see  §  55. 

8.  Find  the  value  of  §  x  |  -*-  2f  +  51  x  ft. 

3r*7x8Ty_, 

*    4AX2X      • 

167 


168  Fractions 

10.  Evaluate  £  +  f  X  f  + 16  X  \  X  8  —  f  X  f. 

11.  What  change,  if  any,  is  made  in  the  value  of  a  fraction 
when: 

(a)  The  numerator  is  multiplied  by  an  integer  ? 

(b)  The  numerator  is  divided  by  an  integer  ? 

(c)  The  denominator  is  multiplied  by  an  integer  ? 

(d)  The  denominator  is  divided  by  an  integer  ? 

(e)  Both  terms  of  a  fraction  are  multiplied  by  the  same 
number  ? 

(/)  Both  terms  of  the  fraction  are  divided  by  the  same 
number  ?     Explain  and  illustrate  each  answer. 

12.  Give  at  sight  answers  to  the  following : 

(a)***,  (c)  |-*-2.  (e).5  +  f 

<&)*  +  . 6.  «|X7.  (/).5xi. 

REDUCTION  OF  FRACTIONS 

263.  The  principles  of  arithmetical  fractions  suggested  in 
example  11  of  the  last  article  will  be  assumed  to  hold  in  alge- 
braic fractions.  In  particular  we  now  assume  the  following 
principle : 

If  the  numerator  and  the  denominator  of  a  fraction  are  divided  by  the 
same  number,  the  value  of  the  fraction  is  not  changed. 

264.  Lowest  Terms  of  a  Fraction.  A  fraction  is  in  its  lowest 
terms  if  the  numerator  and  denominator  have  no  common 
factor. 

265.  To  reduce  a  fraction  to  its  lowest  terms  divide  both  the 
numerator  and  the  denominator  by  all  their  common  factors. 

266.  Cancellation.  Much  time  may  be  saved  in  solving  prob- 
lems involving  fractions  by  canceling  factors  common  to 
both   numerator  and  denominator   if  any  are  present.     The 


Reduction  of  Fractions 


169 


student  should  do  this  at  every  stage  in  the  solution  of  a  prob- 
lem, always  factoring  and  canceling  whenever  possible  and 
never  multiplying  or  dividing  until  all  possible  factors  have 
been  removed  by  cancellation. 

3a 

36     ?.?.?-3     3"  '    ^J(V«     5  c 


1.    rrr 


3. 


(cH^gX^-^^g) 


2a*-2a&4      2  a(a2  +  62)(a-Fl>)(a-^)      2a(a2  +  62) 


267.  In  canceling  a  factor  from  the  numerator  and  the  de- 
nominator of  a  fraction  the  quotient  1  is  not  generally  written. 
It  is  important  to  remember,  however,  that  if  all  factors  of 
either  the  numerator  or  the  denominator  are  canceled  that 
term  of  the  fraction  becomes  1.  If  the  quotient  is  —  1  it 
should  be  written. 

-b  m~*1  1 


Thus, 


3  a2  -  8  ft2      Z(ea^d)  (a  +  b)     3(a  +  b) 
-  1 
b  —  a  -ft— =tt  —  1 


Also, 


a2  -  b1      (a  +  b)  (p^5)     a  +  b 


ORAL  EXERCISE 
268.   Reduce  each  fraction  to  its  lowest  terms : 

1. 


2. 


3. 


X 

7. 

rs 
~r%' 

13. 

—  abc 

xy 

—  a2c 

a2 

a4' 

8. 

Wed 
3c  ' 

14. 

9  %  Vw2 
3uvw* 

ab 

V 

9. 

36  h27c2 
9hk 

15. 

7pq2v2 
21  gv2 ' 

2xy 
4 

10. 

72  mn* 
Smn2 

16. 

-xy2z 

Sc2d 
6cd 

11. 

—  11  a;?/ 
33  a 

17. 

—  a26d2 
4ad 

10  pq2 
5pq 

12. 

14  a 
—  ab 

18. 

102  Mm 
51  &2ra2 

170  Fractions 

EXERCISE 

269    Reduce  to  lowest  terms : 
L     S*£_  l4      x*-l 


13. 


15  xPy*  (x  +  1)2 

147  a3»2  tr         a2 -6a 

15. 


49aV  a2  -la  +  6 

3     27  a^c4  3a26-9a&2 


3a6c  a*-7a2b  +  12a&2 

4         17  a2  -  16  g  -  17 

51a465'  '    a;2  -22  a; +  85' 

35  a^c4  lg      a3&  -  ah* 
42  a^c5'  '    a263-a4&' 

6  39rg*  19      (5a-7)2 
65  rW'  '    50 a2 -98' 

7  (-2a6c)3  2Q         a«  +  afy 

8a62c3    '  '    a?  +  2ajy  +  y2' 

g     (l&tnWpWffin)  21  7a?  +  3 

(14p3)(5m2(/)     *  *    245  a* +  210  a3  +  45  aT 

9     a2  +  ab  22  a4-64 


a  +  a2  a2  +  2  a&  +  62 

10       x2~l  23        g  ~ Xy  +  *  ~*y 
(a-1)2'  '    l-3y  +  3y*-y*' 

w"      3,2  _  y2  ^       0^ +(a  +  6)fl?  +  aft^ 

'    xz  —  yz'  x2  +  (a  +  c)<c  +  ac ' 

12     £-2*  2g     rf+3*  +  2 


3  a2 -12  »2  +  6a  +  5' 

8  a2ft  —  16  ab2  a2n  +  2  an  +  1 


12  a2x  -  48  b2x  26.        .  au  _  1 

27.   Common  factors  may  be  canceled  from  both  terms  of  a 
fraction.     May  common  terms  or  factors  of  common  terms  be 

canceled    in  the    same  way?      Is      2(x  +  y)    =X  +  V?     is 
q»-(s-y)»=-(s  +  y)'?  2+(j/  +  z)     y  +  z 

a?+(p-q2)       (p-q)2   ' 


Reduction  of  Fractions  171 

270.  Algebraic  Signs  in  Fractions.  There  are  three  signs  to 
be  considered  when  determining  the  value  of  a  fraction; 
namely,  the  sign  of  the  numerator,  of  the  denominator,  and  of 
the  fraction  itself. 

Thus,  —  =-  4  but  -  -!£-  =  4.     Also  =^  =  4. 
'  -3  -3  -3 

x  —  2 

271.  Given  the  fraction  — — Find  its  value 

(x  —  4)(a  —  6) 

when  (a)  x  =  0 ;  (b)  x  =  1 ;  (c)  x  =  3 ;  (d)  x  =  5 ;  (e)  x  =  7 ; 
(/)  What  sign  has  the  answer  when  x  is  equal  to  or  greater 
than  7  ?  when  x  is  negative  ?  when  a;  is  less  than  2  ? 

272.  Changes  in  the  Signs  of  the  Numerator  and  the  Denominator 
of  a  Fraction.  By  the  definition  of  a  fraction  and  the  rules  of 
division  we  have : 

(1)  |=2.  (4)  ^2=-2. 

(2)  5|=2.  (5)  _^  =  _(_2)=2. 

(3)  =±  —  %  (6)  -jL  =  -(_2)=2. 

By  comparing  (1)  with  (2),  and  (3)  with  (4),  we  have  the 
following  principle  illustrated : 

1.  If  the  signs  of  both  numerator  and  denominator  of  a  fraction  are 
changed,  the  value  of  the  fraction  is  not  changed. 

A  similar  comparison  of  (1)  with  (5)  and  (6),  and  (2)  with 
(5)  and  (6)  will  illustrate  the  principle : 

2.  If  the  sign  of  the  numerator  or  of  the  denominator  and  the  sign 
before  the  fraction  are  changed,  the  value  of  the  fraction  remains 
unchanged. 

We  thus  have:   *  =  ^  =  -ZlA  = t_=9 

2-2  2  -2 

2  -2        2-2 


172  Fractions 

In  algebraic  symbols,  from  1  and  2  we  have : 

a  __  —  a  _      —  a_        a 
b~~^b~~     ~b~~~^l' 

and  _«  =  _^=-^  =  _?L. 
b  -b        b        -b 

273.  The  student  should  note  carefully  that  changing  the 
signs  of  an  odd  number  of  factors  in  the  numerator  or  in  the 
denominator  of  the  fraction  will  change  the  sign  of  that  term 
of  the  fraction  in  which  the  factors  occur,  and,  therefore,  by 
§  272,  2  the  sign  before  the  fraction  must  be  changed. 

-  If  one  factor  in  the  numerator  and  one  in  the  denominator 
have  their  signs  changed,  which  principle  applies  ?  What  is 
the  effect  on  the  value  of  the  fraction  if  the  signs  of  an  even 
number  of  factors  in  the  numerator  or  in  the  denominator  are 
changed  ? 

Examples 

1.  %TL*:ss-*LzA.    (Why?) 
2-x         x-2     v       J    } 

2.   «=± = h-=* =       (& -«)       =etc. 

(b—c)(c  —  a)      (c  —  b)(c  —  a)      (b  —  c)(a—c) 

Let  the  student  make  other  changes  in  the  signs  of  this  frac- 
tion. 

Would  it  be  possible  to  change  the  signs  of — ^ 

(b  —  c)(c  —  a) 

so  as  to  make  b  -}-  c  a  factor  of  the  denominator  ? 

EXERCISE 

274.  1.    Show  by  multiplication  that 

(x  _  i)(a>  _  2)  =  (1  -  a>)(2  -  x). 

2.  Show  that  (x  -  l)(x  -2)  =  ~(x-  2)(1  -  x). 

3.  Compare  (x  -  l)(x  -  2) (a?  -  3)  with  (1  -  x)(2  -  x)(S  -  x). 


Reduction  of  Fractions  173 

4.   Make  fractions  equivalent  to  each  of  the  following  having 
the  sign  of  the  denominator  changed : 

/  \  a  /j\      —  b  z  \  m 


3  *  '    (-cy  "'        (-a)(-6) 

(b) L.  (e)   - (h)  - 

W        -b  w    -{-by  V  ;   (-a)(-6)2 

«  p*         w  -z^-     (o  p^ 

5.  Give  a  fraction  equivalent  to  each  fraction  in  example  4 
with  the  signs  of  its  numerator  and  denominator  positive. 

6.  If  a,  b,  and  c  are  all  positive  numbers,  what  algebraic 
sign  has  the  number  represented  by  each  fraction  ? 

W   (-  &)c  U;         (_tt-6-c) 

7.  What  change  is  necessary  to  make  the  denominator  the 
same  in  each  of  the  following  pairs  of  fractions  ? 

(a)  z — z>  I — I'        (6) 


a  —  b'b  —  a  (x  —  y)(a  —  b)'    (x  —  y)(b  —  a) 

(c) i , * . 

K  )  (a-2)(a  +  2)'    (2-a)(2+a)' 

8.  Arrange  the  denominators  of  the  following  fractions  in 
descending  powers  of  x  with  the  sign  of  the  first  term  in  each 
denominator  positive. 


(as-2)(3  +  a!)        w2-3*-a?        w        (3-x)3 
(6)  "-»  (d)   uJLdL .         (/) 


(1  —  s)(l+a:)         w       (3 -a;)2  w  '  (2  -  x)*(x  +  2) 


174  Fractions 

275.  An  algebraic  improper  fraction  is  one  whose  numerator 
is  of  the  same  degree  in  some  letter  as  the  denominator,  or  of 
higher  degree  (§  245). 

Thus,  — ^-i and  — - —  are  improper  fractions. 

a2  —  a  —  1  a2  —  1 

276.  A  mixed  expression  is  an  expression  containing  both 
integral  and  fractional  parts. 

Thus,  a2  +  -  is  a  mixed  expression. 
a 

Any  integral  expression  may  be  written  in  fractional  form 
by  supplying  the  denominator  1. 

Thus,  a  +  x  =  ®-±^. 

277.  Improper  fractions  may  be  reduced  to  integral  or  mixed 
expressions  in  the  same  way  as  arithmetical  improper  frac- 
tions, that  is,  by  dividing  the  numerator  by  the  denominator. 

Thus,  ^e  =  aVand«=ii  =  l-i. 
xy  c  c 


ORAL  EXERCISE 

278.    Reduce  the  following  improper  fractions  to  integral  or 
mixed  expressions : 

s2-P 
s  —  t  ' 
ax  +  3 

a 

h  +  k 

h 

m2-4 

• 

11  a  —  m  a  —  b 


"•?• 

5. 

»  1Jr- 

6. 

3     12x\ 
3x 

7. 

a     22  ^ 
4.    — - — • 

8. 

9. 

c  —  d 

c 

10. 

c  —  d 
d 

11. 

abc  —  d 

ab 

12. 

a2  -  ab2 

Reduction  of  Fractions  175 

279.  When  the  numerator  and  the  denominator  are  both  poly- 
nomials, the  problem  is  similar  to  that  of  an  inexact  division 
(§  160).  The  process  will  be  understood  by  studying  an 
example. 

to  a  mixed  expression. 

■   ***&*-•***&   « 

=  x  +  6  -  -I-s       (2) 


Why  is  the  form  (2)  of  the  same  value  as  (1)  ? 
The  division  should  be  continued  until  the  remainder  is  of 
lower  degree  than  the  divisor. 

EXERCISE 

280.   Reduce  the  following  improper  fractions  to  integral  or 
mixed  expressions: 

,     x2  +  1  p    a?  +  bz 

o. 


Change  : 

*.   -j- 

x  —  2 

x2  +  3  x  - 

-  17 

\x-2 

x2  —  2x 

\x  +  5 

bx  — 

17 

5x  — 

10 

x-1 

x2-l 

x-1 

&  +  x  _  13 

x+1 

Sx2  +  lx 

x  +  1 

(x2  -  x)2 

—  X 

a2  +  b2 

a  +  b 

a?  +  b* 

9. 


10. 


11. 


a  —  b 

xh-l 
x-1 

6xy  —  ly*  t 
xy 

^-3^+7 
a?-3 


12 :  as*  +  x2y2  +  2y\ 
x2  +  xy  +  y2 


6.  '       ■  13. 


14. 


273?+  27x2y  +  9a;y2  +  f 
Sx-y 

a?-6a2b  +  12ab2  +  $bz 


a  +  b  a2  +  4a6  +  462 


176  Fractions 

Reduce  to  integral  or  mixed  expressions: 

15.  *<*  +  ?»;  yi.  9±L 


a  +  2b  x  —  1  x  —  1 

w>   3rf  +  5s-8.  18<  _a^.  2Q>      at 


a?  +  x+l  x-1  x  —  1 

281.  To  add  arithmetical  fractions,  we  must  change  them  to 
fractions  having  a  common  denominator. 

In  reducing  fractions  to  a  common  denominator  we  use  the 
following  principle : 

If  the  numerator  and  the  denominator  of  a  fraction  are  multiplied  by 
the  same  number  (not  zero),  the  value  of  the  fraction  is  not  changed. 

ORAL  EXERCISE 

282.  Change  each  of  the  following  fractions  to  an  equivalent 
fraction  whose  numerator  or  denominator  is  as  indicated : 

2x     Ux 


a-l_    ( 
a  +  l      a? 

) 
-1 

a-1      a2 

-1 

1. 


2.   ^  =  LJ.  7. 

b       62  a  +  l       (    ) 

3.  2s.U..  8.  I+!  =  Jl+^, 

5y      15xy  a     b      (     )      (     ) 


4. 


n    =  Smnx^  9    a+l  —  L_l  +  i. 

2n2x       (     )  a        a        a 

«r=y.  io.  3+r5_  =  P-+-2-. 

arx2  b  +  c     6  +  c      &  +  c 

11.  A.+^  =  l±xll. 


a  +  6     a  -  6     a2  -  62     a2  -  62 
(—  a)3     a&       a36        a?b 


13. 1+i+i.LJ+y+t^. 

a     b      c      abc       abc       abc 
14.   What  principle  is  involved  in  all  these  examples  ? 


Reduction  of  Fractions  177 

283.  The   lowest  common   denominator  (L.  C.  D.)  of  two  or 

more    fractions   is    the    lowest    common    multiple    of    their 
denominators. 

Thus,  the  L.  C.  D.  of and is  ab(a2-b2). 

a(a-b)  b(a  +  b)  v  J 

To  change  these  fractions  to  a  common  denominator,  both  terms  of  the 

first  fraction, ,  must  be  multiplied  by  b  (a  +  b)  and  of  the  second 

a(a  —  b) 

fraction,  by  a(a  —  b). 

b(a  +  b) 

Thus,  I = Ka  +  b) and  1 = a(a-b) 

a(a-b)     a6(a-6)(a  +  6)  b(a  +  b)      ab(a-b)(a  +  b) 

284.  To  change  two  or  more  fractions  to  equivalent  fractions  with  a 
common  denominator : 

1.  Factor  the  denominators  into  prime  factors. 

2.  Find  the  lowest  common  denominator  (L.  C.  D.)  of  the  fractions. 

3.  Multiply  the  numerator  and  the  denominator  of  each  fraction  by 
all  the  factors  of  the  common  denominator  except  those  factors  that  are 
in  its  own  denominator. 

The  multiplication  in  part  3  of  the  rule  is  generally  indicated 
in  the  denominator  and  performed  in  the  numerator. 

Why  does  step  3  of  the  rule  not  change  the  value  of  the 
fraction  ? 

Example 

Change  and  to  equivalent  fractions  having 

a2  —  ab  ab  —  b2 

the  L.  C.  D. 

Solution.  a2  —  ab  =a  (a  —  6). 

ab-b2=  b(a-b). 
.-.  L.C.D.  =  ab(a-  b). 

(The  L.  C.  D.  is  the  L.  C.  M.  of  the  denominators.) 
1       =        1        _         b 
a2  —  ab     a(a—b)     ab(a  —  b) 
1  1  a 


ab  -  b2     b(a  -  b)     ab{a  -  b) 


178  Fractions 

EXERCISE 

285.    Change  the  fractions  in  each  example  to  equivalent  frac- 
tions having  the  L.  C.  D. 


1 

1          1 

a2b'    ab2' 
a      b       c 
be'    ca     ab 

X                —  X 

i      1           1 

2 

a2-ab'    ab  +  b2 
K        a             b             c 

3. 

x  —  y      x2  —  y2     y  +■  x 
a         a             b 

x     y    y  — x 

(-yY  {-yf 

Hint.    In  example  6  change  the  denominator  of  the  second  fraction  to 
x  —  y.     Explain. 

7  1  11 


8. 


a? +  4^+3'    aj2_1'    x+1 

1  1  1 

tt3_53'      a2  +  a6  +  62'      a2_52" 


Q  x+y  y  +  z  z  +  x 

(y-z)(z-xY  (z-x)(x-y)'  (z-y)(y-x)  . 

Solution.     First  change   the  signs  in  these  fractions,  to  avoid  the 
repetition  of  a  factor  with  opposite  signs. 

x  +  y  y+z  z  +  x 

(y  — *)(#  —  £)'    (z-x)(x-y)'  (y-z)(x-y) 
Why  is  the  change  made  in  the  last  fraction  permissible  ? 
L.C.D.  =  {x-y){y-z){z-x): 

x  +  y x2  —  y2 

(y  -  z)  (z  -  x)  ~  (x  -  y)  (y  -  z)  (z  -  x) ' 

y  +  z y2  —  z2 

(z  _  x) (x  -  y)  ~  (x  -  y){y  -z)(z-xY 


10. 
11. 


( y  -  z)(x  -y)  (x-  y)  (y  -  z){z-  x) 

1  1              1 

a2-6a  +  9'  9-a2'  a-3* 

1  1                      1 


a;2- a; -12'  x2  +  8x+15'  z2  +  x-20 


Addition  and  Subtraction  of  Fractions        179 

3a  +  b  a  —  b 


i/G. 

6a2-a&-562'  18a2  +  21  ab  +  5  b* 

13. 

a  +  x    a  —  x    a2  -+-  x2      4  ax 

a  —  x'  a  +  a'  a2  —  x2'  a2  +  x2 

14. 

5  a    16  a2  —  17  ab         b 

6  6'    12a6-662  '  b  -2a 

15. 

3              2         2  a  + 15 
2a-3'  3  +  2a'  4a2  +  9* 

1« 

b  -  a               2ab  +  c2 

c-2b    bc-2b2-ac  +  2a? 


ADDITION   AND   SUBTRACTION   OF   FRACTIONS 

286.  In  arithmetic,  only  the  same  kind  of  units,  or  the  same 
parts  of  units,  can  be  added.  Hence,  if  two  or  more  fractions 
are  to  be  added,  unless  they  already  have  a  common  denom- 
inator, it  is  necessary  to  reduce  them  to  equivalent  fractions 
having  a  common  denominator,  before  they  can  be  added. 
Their  sum  is  then  found  by  adding  the  numerators  of  the 
fractions  and  dividing  the  result  by  the  common  denominator. 

Thus,  I  +  f +  f  =  I±|±3  =  S 

3      12      15     60      60     60  60  60        ™ 

287.  The  same  principle  applies  when  the  difference  of  two 
fractions  is  to  be  found.  The  difference  of  two  fractions 
having  the  same  denominators  is  the  difference  of  their  nu- 
merators divided  by  their  common  denominator. 


'hus,  -- 
'  8 

5_7-5_2_l 

~8         8         8     4' 

and  — 
12 

7  _  15      14  _  1 
18     36     36     36 

180  Fractions 

288.   The  sum  or  the  difference  of  algebraic  fractions  can 
be  found  in  the  same  way. 

1     a  i   ^       c  _a  +  b  +  c 
m     m     m  m 

2.    Add:  -J5-  +  -^+_J^. 
2  an2     2  bm2     abmn 

Solution.  L.  C.  D.  =  2  abm2n2. 

m         bm2  •  m  bmz 


2  an2     2  abm2n2     2  abm2n* 
n  an2  •  n  an* 


+ 


2  6m2  2abm2n2  2  abm2ri2 
p     _  2  mn  •  p  _    2  mnp 

abmn  2  abm2n2  2  abm2n2 
p     _  bm3  +  an8  +  2  mnp 


2  an2     2  6m2     «6mw  2  abm2n2 

3        a  & 

a  —  b     a  +  b 
Solution. 

L.  C.  D.  =  (a  -  6) (a  +  6)  or  a2  -  b2. 

a  b     =a(a  +  b)     b(a  -  b)  =  (a2  +  ab)-(ab  -b2) 

a-b     a  +  b       a2  -  b2        a2  -  b2  a2  -  b2 

=  a2  +  ab  -  ab  +  b2  _  a2  +  b2 

a2-b2  a2-b2' 

Check.     Put  a  —  2,  b  —  1. 

Then-^ L-.£±£s«i     *«#,«•«§. 

2  -  1      2  +  1      22  -  l2  3     3'        33 

Why  do  we  not  put  both  a  and  6  =  1? 

289.     To  add  or  subtract  fractions : 

1.  Reduce  the  fractions,  if  necessary,  to  equivalent  fractions  having 
the  L.  C  D. 

2.  For  the  numerator  of  the  result,  write  the  numerators  (in  paren- 
theses if  they  are  polynomials),  joined  by  the  signs  between  the  frac- 
tions; and  for  the  denominator  of  the  result  write  the  L.  C.  D. 

3.  Remove  the  parentheses  in  the  numerator,  and  collect  the  terms. 

4.  Reduce  the  result  to  its  lowest  terms. 


Addition  and  Subtraction  of  Fractions        181 


Example 
1  a 


—  9 


a  —  b     a  +  b     a2—  b2 

Solution.        L.  C.  D.  =  (a  +  ft)  (a  —  ft  ). 

1 1     '         a  a  +  ft q  —  ft .       a 

a-fc     a+6     a2  -  ft2  ~  (a +  &)(«- ft)      (a  +  6)(a  -  ft)     a2- ft2 

_(a  +  ft)-(a-ft)  +  a 
(a  +  6)(a^ft) 

a  +  2& 


(a 

+  &)(«- 

Check. 

Put 

a  =  2, 

6  =  1. 

Then 

1 

1 

,      2 

2  +  2 

'  2 

-1 

2  +  1 

1  4-1 

3.1' 

1 

-f  +  * 

=  f;or| 

*) 


or  l-i  +  f=f;or|=f. 

290.  The  fraction  line  may  be  thought  of  as  having  the 
same  effect  on  the  numerator  of  the  reduced  fraction  as  a 
parenthesis.  In  writing  the  numerators  over  the  L.  C.  D. 
the  parenthesis  is  used  to  indicate  that  the  whole  numerator 
is  to  be  added  or  subtracted. 

ORAL  EXERCISE 

291.  Perform  the  indicated  operations: 

1.    |  +  f.  6.    5- 

2  +  5 


3.     -  +  ?• 

y    y 

ft2     6 

5.  J-+-L_. 

a  +  6      a  —  6  a  +  ft 


a; 

2~ 

a; 
3' 

a 

2 

a 
5* 

.T 

5 

2/ 

2/ 

a 

6s 

c 
"ft3 

1 

182  Fractions 

EXERCISE 

292.    Perform  the  indicated  operations : 

~-r-~_t  q     2x     5y     x     3y 

8*    15  +  12+5~T* 

Q     5a     26,3 
a;2      «?/      y 

10.    l^+7a»      17  rf 


2. 


a  +  26 

a—  b 

4        ' 

2 

2crZ      c 
a&      a 

1 

2 

a-f  b     a 

-b 

a-2b 

2a-5b 

5 

5 

a;      a;  — 4 
4         3 

x  —  5 
6 

2^  +  y  , 

x  +  22/ 

3        ' 

3 

7       ,10 
12*  +  21 

1         1 

X X X- 

7        4 

4.  „ "  •  11. 


5.    =■  —  =-- h- -•  12. 


13. 


7.   —X+  —  X  —  ~X  —  Ta>  14. 

2 


9  #?/      12  xy      18  #?/ 

a(x  +  a)      aj(a;  +  a) 

3a  +  5      2a  +  5 
a  —  b         a  +  b 

a-\-b      a  —  b 
a  —  b     a  +  b 

5  7 


4  a;  —  4      6  a; +  6 


15.    -^— + 


1  +  a;     1  —  x     1  -\-  x 


16.    -i_  + 


2z-3     3-2#     2ic2-»-3 

17      a?~4       3a;-5  .5a?  +  9a;  +  14 
2a>-l       a  +  2        2a?  +  3a;-2 

18.    -A- f-i 8_  +  3*  +  7 


19. 


x-  4-  as      a;2 

8a2 


x-\-2a     x—  2  a      a3  —  4  a2# 


20.  3<?    p2  +  r/2 1^  +  ^-3, 

293.  By  supplying  the  denominator  1,  we  may  include  the 
reduction  of  mixed  expressions  to  fractional  forms  under  the 
addition  and  subtraction  of  fractions. 


Addition  and  Subtraction  of  Fractions        183 

EXERCISE 
294.    Perform  the  indicated  operations : 
**         m  .      m2 


1.    ra  + 


m  —  1      1      m  —  1 


m(m  —  1)  . r; 

m  —  1         m 
m2,  —  m  +  w2 

m  —  1 
2ro2-  m 


a2-62      a  +  &  p-<y       Vjr      ^ 

3       262        0         6  2a  +  l      3a-l      a(a  +  3) 

'    a2-62        "**<*_-&'  '     a  +  l        1-a         a2-l   ' 

Hint.     Change  the  second  fraction  to  an  equivalent  fraction  whose 
denominator  is  a  —  1. 

a  1  4— #2  a2 


62      6- a  2a+a2       2a       4+2x 


8. 


2a2  +  2a;-19     2a2-25 
a?  +  3  3  +  a; 

9    (p-q)2  +  (p  +  q)2     p      q  , 

4p<?  4^         2(/      2j? 


«. 


Hint.     Remove  parentheses  and  collect  terms  before  changing  to  a 
common  denominator. 

11  3a;  4  2y     2x  —  ky      x  +  Vly  ^ 
Sx  —  Sy     kx  —  ky      6y  —  6x 

12  3 ft  +  4 y  ,  2x—  5y _  6a?  +  5y 

3»4-y       3i/  +  9a;      12a?  +  4^" 

13  s-2y     2      6#-2ag 

a-6?/     3      3a?-182/' 


184  Fractions 

Perform  the  indicated,  operations : 

14      3a2  +  62        2a  +  b        a-2b 


16. 


la* -4b2     6b -6a  6a  +  6b 

3x        .    x2  +  5.Ty  2 x2  +  a% 

4  ?/  —  4  a;     4  a2  —  4  ?/2  2  a;?/ +  2? 

a2-3a&      6g3  +  a2&  3ft2 

2a2  +  b2      2a2b  +  b*  ab-b2' 


1?     5a;      10^-17^  |        y 
6y       12  xy  +  6y2      2x  —  y 


18. 


19. 


9a;2  +  2a;y       '     3  a;2    *         11  x2  +  xy 
12xy  —  6y2     4xy  +  2y2     12  x2  —  3  ?/2 

a; 4a;y 

2x  —  y     4:X2-\-4:xy  —  Sy2 


20.  21a  +  116-(7a  +  66>2- 

46 

21.  »2      +^! °L-. 

02  +  a;2      a2  —  aa;     a;2  —  a2 

22.     y    +  50a;2  +  5x 

y  -+-  5  x     25x2  —  y2     y  —  5  x 
23     ^1       4        1  » 


24. 


(a  +  a;)2     a2  —  x2      (x  —  a)2 

5a-2       3a  +  l       2a2-5a  +  7 
24a- 6     36a +  9         48a2 -3 


25.      *  +        *  *  ' 


26. 
27. 


2a;     8-4a;      8  +  4a;     a;2  -  4 

1  1  1 1_ 

2a;-f-a;2     2x     2a;  +  4     a;-2 

a;2  +  xy  +  j/2 1 

x4  +  4  ?/4         a;2  -f  2  a;y  +  2y2' 


28.        «  +  *  a;  -  1  2 

a;2  +  a;  +  1      a;2  -  a;  +  1      a;4  +  a;2  +  1 


Addition  and  Subtraction  of  Fractions        185 


a2_2g  +  3  a- 3  1 

a3  +  1  a2  -  a  +  1      a  +  1 

30. 


6a2  +  a  —  2      3  a2 -a- 2 

si.  l£±£+    3         4 


32. 


2a;2 -8     a; -2      2  -  a; 
2  2  36 


a-  b     a  +  b      a2  +  ab-2b* 

x2-5  x-2      3 

a;2-12a;  +  27      x2  -  9 


34     l  +  3x  9-lla;      u(2a;-3)2, 

*    5  + 7a;  5  -7x           25 -49a;2 

x  1/  7a;        5x2  -f  2xy        2a%/   \ 

2a; +  2y  2^/  -  x         x2  -  y2        xy  +  y2) 

36.    l_f^l_JlL__M 
\x  +  y      x2  —  y2 


37.    ^ --f 


a;2  +  2/2 
5 


a;2  +  3a;  +  2      a-2 +  5  a; +  6      a;2 +  4  a; +  3 


2x2-7x  +  6     2a;2-a--6     9-4X2 

39.     ,    ~2      +        W,+  1      .+       * 


(m  —  l)2  wi2  —  2  m  +  1      1  —  ra 

40         «2       |  &2          «2  +  &2 

a&  +  b2  ab  +  a2         ab 

41.    z ~ z+:  b                           C 


(a-6)(a-c)      (6-c)(6-a)      (c-a)(c-b) 
42.    ^^1 r+..  *  .+  * 


(a  —  b)(a  —  c)  (6  —  c)(6  —  a)  (c  —  a)(c  —  &) 

43  a2  —  be  b2  —  ca         .        c2  —  a6 

(a  +  b)(a  +  c)  (6  +  c)(6  +  a)  (c  +  a)(c  +  6) 

4 .  a  +  6 ?>  +  c  c  +  a 


(c  —  a)(c  -  6)      (a  —  6)(a  —  c)      (b  —  c)(b  —  a) 


186  Fractions 

MULTIPLICATION  OF   FRACTIONS 

295.  The  product  of  two  or  more  arithmetical  fractions  is 
the  product  of  their  numerators  divided  by  the  product  of 
their  denominators.  The  result  should  be  reduced  to  lowest 
terms. 

Thus,  1.    2x5  =  Lx_5  =  l_0.  3X|  =  3 

373x7      21  £     2J      7  • 

7 

296.  The  product  of  two  or  more  algebraic  fractions  can  be 
found  in  the  same  way. 

a      c  _ac 
bXd~bd' 
To  shorten  the  work,  we  usually  cancel  all  factors  common 
to  the  numerators  and  denominators  before  multiplying. 
a         4 

'    fipc     Wfe      5  c2 
5 

0     a2  —  x2      a2  -  b2      (_  ,     ax  \ 


a  +  b      ax 

-f-  a;2     \        a  —  x) 

_  Cfl--^0  (jUt^c)      (a  -  b)  Cet^r-tf)        a2 

Mlr^^O 

2(a-Mr)          -a — ar 

_a2(a-b) 

X 

ORAL  EXERCISE 

297.   Multiply  the  following : 

1        2    .  O                         a       2   .   1 

3.    }  .3.                  4.    K-i). 

a 
5.    -.0. 

a-b     a2-b2t 
a  +  b          2 

6.    *  •  ^. 

o    a  +  6        * 

62        6 

a  -  6     a2  -  b2 

'■(-©(-!>  -H)-(A) 


Multiplication  of  Fractions  187 


12     (a~b\2       a  +  h  14     2a%3?       3b* 


a  +  b)      a2-b2  3  68       4c4a; 

15.  What  is  the  numerator  of  the  product  when  all  the 
factors  of  the  numerator  are  canceled? 

16.  What  is  the  nature  of  the  product  when  all  the  factors 
of  the  denominator  are  canceled  ? 

17.  What  is  the  product  if  all  factors  of  both  numerator 
and  denominator  are  canceled  ? 

298.    To  multiply  expressions  some  or  all  of  which  contain  fractions : 

1.  Reduce  all  integral  and  mixed  expressions  to  the  fractional  form. 

2.  Factor  all  polynomials  that  can  be  factored  in  the  numerators  and 
denominators. 

3.  Cancel  all  factors  common  to  the  numerators  and  denominators. 

4.  Multiply  together  the  remaining  factors  in  the  numerators  for  the 
numerator  of  the  result ;  and  the  remaining  factors  in  the  denominator 
for  the  denominator  of  the  result. 


1. 


3 
4 

Examples 

_3c2 
"   8  ' 

\a?~ 

iYi+M 

J\           XX 

vU3- 

i) 

1- 

X2      X2  - 

f  x  +  1 

X2 

Xs 

x8-! 

(Changing  to  fractional  form.) 

-1 

_(l  +  aOO^ 

X 

3 

(x^T)(x2>^fl) 

(Factoring  and 
canceling.) 

-1 

"xor 

1+  X 

188  .    Fractions 

EXERCISE 
299.     Perform  the  indicated  operations : 


l 


xfx^.  s    —-  —  .  ^ 


2.    7f  x  ^-.  &y    V*z     abc 

3-    (f  +  iXA  +  A)-  9     (3ct*6)»  ,(5c)*       (4  6)* 

4.  5  mn  .  lM .  (5  c^)3  '  (6  «)5  '  40  «3&5c' 

5mn2  f_3^_2Y    /     2aV\s 

5.  13m2^.^^-2.  'V      4  6c3;     V       *** 

8ti2 

a2-62 

a  KA,  U  K,  J.X. XI — 

6-  Yc'w'ab  a~b    \    «2+2«6+62 


a  — 6     V     a 

12    wM^H2(    (m  —  y)2 
W2  —  V2        u2—uv  +  v2 


'■  (-H)c-exf> 

13  2ab  ,2q  +  36 

4  a2  +  12  a6  +  9  62  '     a  -  6 

14  a+5        .  a;2  +  8a; +  15, 
a;2  —  a;  —  12  '         x  —  4 

17.     a3-&3     l      g  +  6  24     frjV+% 


(a +  6)2    2    a2-62 


\y    ay 


18.    2-   m2~P2      ^3-/>3,  25.       1        /I      IV 
3    (m-p)2     m+p  x  —  y    \x     y) 

i9.  Aa_i6y.  i  ^|/i+iY-s 

\4        2  J     a -2b  2\a     bj\a- 

20.  _JL_.(a_6).  27.    ^a-3Y 77T ss\ 

a2-b2    K         J  \2         J\a2-12a+36j 

21.  ^-lY^  +  A  28.    a!±^+A2.«!±l3. 
U        A«       /  a*-a&  +  62    a3  -  63 


Multiplication  of  Fractions  189 

29  (*2±yl-x\-JL^.t=£. 

V  y       Jx-y  «3+2/3 

so.  (x+y+Q^n\(?-y). 

V  x         y  )\y    xj 

31.  (i.-9y.f   1l  \ 

32.    kl-^V +-+I      \ 

\'2y     3xJ{     3<*  +  S#-2f) 

\a      oj  a  -\-o       \     x  —  yj 

35.  (*+*yjL_y, 

\y      xj\x  +  yj 
„    a4  -  64    a2  -  52    a6  -  ft6 

OD.      •  ■ *   • 

a3-63    a3  +  63    a2-f&2 
37     «2-(&-c)2    Ca  -  by  -  c2 

C2_(5_a)2*    (5_c)2_a2* 

»  (^)'(^r> 

2x*-8x+6  ^  x*  -  9  x  +  20  §  a?2-7a?  r 
a;2  -  5  a;  +  4  '  a2  -  10  x  +  21  '  2  a2  -  7a?' 

4  a      /        IV 

-J  x 


40. 
41. 

42. 


(z2-l)2V 
9x*-6x       2z2  +  3z-9    2<c2  +  13o;-7 


4a;2-8z  +  3    6x2-7a  +  2         2a?2  +  6a; 

-  (-SfcVJ- 

44    1-aO   g»-l  /I  |      »   > 

1  +  2/     a;  +  a2    ^       1-x^ 


190  Fractions 

DIVISION    OF   FRACTIONS 

300.  The  reciprocal  of  a  number  is  1  divided  by  the  number. 

Thus,  |  is  the  reciprocal  of  2  ;  f  is  the  reciprocal  of  § ,  for  1  —  |  =  f . 
The  reciprocal  of  a  fraction  is  evidently  the  fraction  inverted. 

301.  Division  has  been  denned  as  the  process  of  finding  one 
of  two  factors  when  their  product  and  the  other  factor  are 
given. 

Thus,  |  X  | « If,  hence  ||  -  |  =  J  and  \ f  -  f  =  f . 

1.  Divide  if  by  -f. 

1.  Let  £|  -f-  f  =  q.     (A  quotient.) 

2.  Then  if  =  $  x  q.     (By  definition  of  division.) 

3.  .'.  *xii  =  Sx$.g  =  ff.     (Why?) 

3 
4    I^5  =  ^      J  =  3 
28  "  7      2£      £     4 
4 
(From  1  and  8,  since  each  =  <?.) 

2.  Divide  -  by-. 

1.  Utf+£** 

2.  Then  ?=cxg.     (Why  ?) 

b     d 

3.  ..Jx^^x^?  =  2.     (Why?) 

c      b      c     d 

4     2  -i.^  —  §.  v  ^  =  ^. 
'    b      d     c      b     be 

(From  1  and  3. ) 

302.  Step  4  in  each  of  the  examples  of  §  301  gives  us  the 
usual  rule : 

To  divide  one  fraction  by  another,  multiply  the  dividend  by  the 
reciprocal  of  the  divisor. 


Division  of  Fractions  191 

In  case  the  dividend  or  the  divisor  is  an  integral  or  a  mixed 
expression,  or  is  the  sum  of  two  or  more  fractions,  it  must  be 
changed  to  a  single  fraction  before  the  rule  is  applied. 

This  method  of  division  may  be  used  also  in  dividing  a 
fraction  by  an  integer,  since  any  integer  can  be  written  in 
the  form  of  a  fraction  by  supplying  1  as  a  denominator. 

ORAL  EXERCISE 
303.   Find  the  quotients : 

1.     2     •     3-  11.     1  -t--- 


2.    l-h2. 

A     a      b 

4.  — ! — • 

b      a 

k      a  .   K 

5.  -  +  fc 


12.    1 


i-h) 


62  :  Yd 


HM-i> 


-  BH9' 

10.    (-«)2-^- 

20.  What  is  the  reciprocal  of  - 

b  x 

21.  Is  -  +  -  the  reciprocal  of  -  +  -?  If  not,  give  the  cor- 

a     c  b      d 

rect  reciprocal. 


13. 

b 

--rC. 
C 

14. 

ab  s • 

ab 

15. 

2  be      n 
-—-J-  5c. 
3a 

16. 

1 

a  -. 

b 

17. 

,        ab 

abc-i 

cd 

18. 

i    .  m 
n 

19. 

15ab  .   1 
cd        cd 

? 

ofa  +  6?     o{aj? 

192  Fractions 


EXERCISE 


304.    1.   What  is  the  reciprocal  of  -  +  1?     of  a  +  6?    of 


x 

2.  Find  the  reciprocal  of  3^- ;  of  a  -\ 

3.  Find  the  reciprocal  of  -  +  - ;   of  a  +  2  + i. 

x     y  a 

Find  the  quotients : 

*  6*      c»      c3 

5     «^^!.  9     27a-2y  .  12 x* 

3  '   2*  "     25z3    '  52/z2' 

6.    2i2^3.  10.    lot* +  3* 


6  63        64  3 

7     2a?^_3abt  n     f±a 


J  A2/J' 


6c         c2  V9e/^    4 

12.  Divide  — ~^L  by  the  reciprocal  of     ™    • 

13.  The  product  is  —  and  one  of  the  factors  is  — — .     Find 
the  other  factor.  4  c  8  &d 

2  a  14  a2 

14.  The  quotient  is  —  and  the  dividend  is  — — .     What  is 

the  divisor  ? 

15.  (a-lWl-M.  is.  (^a-i6)+(4a-86). 

16.  a'-ft'^    «-&   .  i9.    (aJ»-6a!-7)-l>-7). 


(1? 

-62 

a 

-6 

a3 

+  63 

(a 

+  &)2 

x2 

-2x 

-3 

cc  - 

-3 

a2 -4a         a2  -  16  V  T  *;     \^5     l<y 

21.   4-(-i)-5^(-|)-7^(-«). 

3a?  +  5  .  9a2  +  30a;4-25 
'    6a-9  '         4x2-9 


Division  of  Fractions  193 

„    (2a     3b\  .  (4a?  +  9b-     ,\ 
25-  ^~2^j""i"^2^ 7 

Note.  In  a  series  of  multiplications  and  divisions,  the  divisions  may 
be  changed  to  multiplications  by  inverting  the  fractions  immediately  pre- 
ceded by  the  sign  of  division. 

Thus  -  ^--\c  —  —  —   —   -  =  a^e 

'  b   "  d     f      b'  c'f~  bcf' 

86.    |§  +  |5x52.  27.    f#+i^»W.  +  i\ 

5cd*      5d       a  \        *       /     \       */ 

/g2 3  #  _l  2 

28.  By  what  must  - — —  be  divided  to  get  as  quotient 

x-2?  *°-7*  +  12 

x  —  4 

29.  By  what  must  — — — x  "*"       be  multiplied  to  get  as 

"L      o  2z2  +  3a;-2  F  5 

product  £=_£? 
x  -{-  2 

30.  What  is  the  dividend  if  both  divisor  and  quotient  are 

x-1' 

a-3b  a  +  3b 


31. 
32. 


a2-2ab-15b2  '  a2  -  8 ab  +  15 62 
1  1 

a2  —  4 a; +  4  '  8  —  a?" 


33.  f,_3  +  -^-V--f^-l+-^\ 
V  2a-6y     2       V  a  -  3J 

34.  f«  +  5+£V/r*_5  +  £\     35.    a  +  b  .  a2  +  2ab  +  V 


b      c     a  J     \b      c      a  J  a  —  b  a2—b2 

36.  ^_23,2+^W--MY 
[  3  *        a?J     \x*y      x     SJ 

37.  Find  the  reciprocal  of  —  —  ^- 


194  Fractions 


COMPLEX  FRACTIONS 

305.  A  fraction  whose  numerator,  or  denominator,  or  both, 
contain  fractions  or  mixed  expressions,  is  a  complex  fraction. 

2  „. 1         1.1 

3  a         a     b 

Thus,  '■= ,  — - —  and =•  are  complex  fractions. 

7  a~b 

306.  Inasmuch  as  a  fraction  is  an  indicated  division,  a 
complex  fraction  may  evidently  be  simplified  by  dividing  its 
numerator  by  its  denominator.  In  the  above  examples  we 
simplify  as  follows : 

T 

2.  a_a2-f-l  .  a2  +  l  -l_a2  +  lj 

6  a  a        b~     ab 


3. 


M 

a     &     6  +  a     6  —  a     6  +  a      «&     _  &  -f-  a 

1_1_~     a&  a&  ab      b  —  a~  b  —  a 

a      b 


307.  To  simplify  a  complex  fraction,  first  change  the  numerator  and 
the  denominator  each  to  a  single  fraction  and  then  multiply  the  numera- 
tor by  the  reciprocal  of  the  denominator. 


Example 


1_2  j     l-2a?  +  a;2 

sc2      #         __  x2 

~Z  2~~  a2  +  a-2 
x  +  1 


X  X 


x-1 

'x^ft* . .  as  x-1 


X 


a?  (£>"*)(x  +  2)      aj(a>  +  2) 

x 


Complex  Fractions  195 


Care  should  be  taken  in  such  complex  fractions  as  -  to  in- 
dicate which  line  separates  the  numerator  from  the  denomi- 
nator.    This  may  be  done  by  using  a  heavier  line. 

a 

b      a      ,  .,    a      ac 


EXERCISE 
308.    Simplify  the  following  : 


a  +  b  .  a 


1.    -""*■  *±-±^  + 

2  +  i-  q-6      q+6 

i+i  2(a2  +  62) 
2      _2_ZU_. 

~^T7±  ab-2cd 


2 f"1 


3. 


y  +  ^  «262  -  4  c2d2 

aHjf  f-<? 

y     x  x2  +  xy  +  y* 

\x  -y 


4.    l-\ — -.  io. 

l-x  4x2-9y2 


1-a 


1+P- 


11. 


1-p 


1~T±~  ^  +  i 

1  +  a  pi^tf 

x_±l  +  1  1        2mn 


6.     ±Z* 12  *'  + 


a?  +  1 ^  ra  -f  ?i 

*  — 1  3 

a         1-a  12       3 


1  +  a         a  y2     sy     & 

_a l_-a'  ld*  1_3       ' 

1  +  a         a  w     a; 


196  Fractions 

Simplify  the  following  : 

1  -u1  j       2    A.A            „„            « 
M     a^  +  aT~6U  +  *J  2a    ~ ' 

14       f+2+*      '        6+cTl 


a  +  x     a  —  x 


15.  »-'      a  +  X.                                    [bj-1 
a  +  x  ,  a  —  x  21.    -$-$ ■ 

a  —  x      a  -f-  x 

16.  Find  the  value  of  the 

fraction  in  exercise  15,  when  9        5  6     3  52 

a  =  2  x.  mkmt  a        a3 

22. 


17. 


18. 


19. 


p  —  q     p  -\-  q  2  a? 

1  1  "' 

n»    y     2    & 

ax  23. 

6      # 


a    a; 
3 


24. 


2/ 
cc  +  2 

2j/ 
a; 

2 

2 

I 

X 

+  2 

4  + 


5  +  i 


25. 


5  +  2 

2xy     Sf 
3          4 

JL_A 

3?/     4# 

5a_8j/    *  _5^       8 
6       9        6yz     9xz 


XL  EQUATIONS  CONTAINING  FRACTIONS 

309.  To  solve  an  equation  containing  fractions,  we  first 
change  the  equation  to  one  not  containing  fractions.  This 
process  is  called  clearing  of  fractions.  The  process  will  be 
understood  by  an  example. 

Solve -i-  +  ?  =  JL+*L 

Ix     x     2x     28 

Solution.  J_  +  §  =  JL+JL. 

Ix     x     2x     28 

Multiply  both  members  of  this  equation  by  28  x,  the  L.C.D.  of  the 
fractions.  The  resulting  equation  will  not  contain  any  fractions  ;  that 
is,  it  will  be  cleared  of  fractions. 

4  14 

4  +84  =  70  +  9s.     (Why?) 
-9x  =  70-  4-  84.     (Why?) 
-9x=-18.     (Why?) 
x  =  2.     (Why  ?) 

Check.    Substitute  2  for  x. 

ff*=fforff 

In  practice  the  student  will  generally  find  it  possible  to  omit  the  second 
equation,  canceling  mentally. 

310.  To  solve  fractional  equations: 

1.  Clear  the  equation  of  fractions  by  multiplying  every  term  of  both 
members  by  the  L.  C.  D.  of  the  fractions. 

2.  Solve  the  resulting  integral  equation  in  the  usual  manner. 

197 


198 


Equations  Containing  Fractions 


ORAL  EXERCISE 


311. 


Solve  -=7; 

X 


XX  XX 


oQl       3      2       k     4       Q     1  .  1      1 

2.  Solve =5;— =  8;  -  +  -  =  -• 

xx  ox  Z     4:     x 

3.  How  do  you  multiply  a  fraction  by  an  integer  ? 

4.  Give  in  order  the  principles  used  in  solving  the  equa- 
tion in  §  309. 

5.  Explain  transposing. 

6.  How  do  you  rind  the  L.C.D.  of  several  fractions  ? 

7.  Why  does  multiplying  the  several  fractions  of  an 
equation  by  their  lowest  common  denominator  clear  the  equa- 
tion of  fractions  ? 


EXERCISE 

12 

.     Solve  the  following  equations: 

l. 

x  .x _5 
2      3~6* 

9. 

1  +  1  =  1. 

2  3     & 

2. 
3. 

A  +  3  =  18. 
m 

Sp 

10. 
11. 
12. 

2|tf  =  5. 

i^  =  _5. 

—  a 

3  a? -fa?  =  6* -22. 

4. 
5. 

%x  +  \x=10. 

2x  =  ' 

13. 
14. 

!-12  +  S. 

a;              a; 

a     a     a      a  —  32 
2     3     4      12 

6. 

X           4:X            K 

2~T  =  5- 

15. 

5=15-1. 

2/          y 

7. 

15  =  7. 

t 

2c     3c     c 
5        2       2 

16. 

My-  2.25  y  =  3.5. 

8. 

32. 

Solution        .5  y  —  2.25  y  =  3.5. 
-  1.75  y  =  3.5. 

y  =-2. 

17. 

—  +  5=13. 

—  X 

20. 

A.5x-lU 

18. 

.lx  —  .1  =  ,5x- 

-5.1. 

21. 

-12  +  13 

19. 

i5-5  =  21. 

X 

22. 

T 

.25  m  +  3.8 

Equations  Containing  Fractions  199 


25. 


313.     The  following  examples  will  show  the  arrangement  of 
the  work  in  the  solution  of  fractional  equations : 

!.    Solve x-^*- *=^  =  ?L=lZ. 
5  4  2 

Solution.  L.  C.  D.  =  20. 

4(x  -2)  —  5(x—  3)  =  10(x  -  7),  (Multiplying  both  mem- 
bers of  the  equation  by  the  L.  C.  D.) 
4x-  8-  5x  +  15  =  lOx-  70.    (Why?) 
4x-5x-10x  =  -70  +  8-16.     (Why?) 
_llx=-77.     (Why?) 
x  =  7.     (Why  ?) 

Check.  L^?  _  Ij^§  =  L^_Z  0r  1  -  1  =  0. 

5  4  2 


2.   Solve 


8  —  x     3  #  —  5_#  +  6_sc 
6  ~3      :        2         3 


Solution.  L.  C.  D.  =  6. 

(8  _  x)  +  2(3x  -  5)  =  3(x  +  6)-  2x.     (Why?) 
8-x  +  6x-10  =  3x  +  18-2x. 
-x  +  6x-3x  +  2x=18-8  +  10. 
4x  =  20. 
x=  5. 

Check.     8^-5  +  15-_5  =  5+_6  _  5         8  +  10=11_6 ,01.23  =  23 
6  3  2         3'       6       3       2       3  6       6 

Note.  Treat  the  numerator  of  a  fraction  exactly  as  if  written  in  a 
parenthesis,  as  is  done  in  the  second  step  of  each  example.  Do  this  in 
every  case  and  use  great  care  in  removing  the  parentheses.  See  §§  100 
and  102. 


2  a? 

5x      ., 

4     5 

1. 

1,2,3 

--tt  =  1- 

2. 

3. 

-+-  +  - 

3 

9 

a;      a; 

XXX 

200  Equations  Containing  Fractions 

EXERCISE 

314.     Solve  the  following  equations : 

■i         o        o 

Note.  It  will  generally  shorten  the  work  of  solving  an  equation  if  all 
fractions  having  the  same  denominator  are  combined  before  the  equation 
is  cleared  of  fractions. 

4.    J— 1  =  A.  5.    i2_l+J_=20i 

6  a;     5  a;     10  a;      a;     3  a? 

9a;     12a;      8a;     24a;     72 
7.    J»+Ji-e«it  9    8ar--  =  — +153. 

g    8(2  +  5r)  =  9r+2  4     10 

9  2  10.   ^-(i+lf)=H(3*+l). 

11.   i(5x  +  l)-i(±x+5)=\(3x-l)-^(6x  +  4). 

12.30(2-0-1  =  1^-^. 

16  2     ^     6         9V  ; 

14     7      l  =  23-a?       7        1 
a;     3         Sx         12     4a;* 
8      a;-4      -x     /2-16a;     2  -  xy 

15.     — 


--(" 


9        11  3       V      33  9 


16.  A4|+^±l-*±^V  =  9a!  +  19. 

17.  («_£)* -(« +4 


17.  (B-^-^+jy.* 

17_a^ 
9       16 
x-1 


19.   2  a; 


3 


-\5x 
J      4' 


20.   3a?_3a?-19_8==23-g  +  5a?-38  +  ia 


21. 


Equations  Containing  Fractions  201 

4*-l     3a;  +  5     (x  -  4     3 


3  4 


/a; -4      3\ 
V     6         4/ 


22     a^-3      x-25==7      2  +  x 
'        7  5  5 

23.    I^_!(a;  +  3)  =  !(z  +  2)-6. 

-.      /3a?-l  .  2a?  +  l\     5-2a?      7a?-l 
^4_h3y  3  8 

25.  3ft-2*  +  5  =  16-7a;  +  19-2a;  +  1. 

7  2  3 

26.  if(z-  4)  =i(8z  +  B). 

L_     5  x  -  .4  .  1.3  -  3  x  _  1.8  -  8  x 
27'    ~3~  +        2~~ "      1.2      ' 

2-4z      3z-4      a;-6 


28.    j 
29. 


3  2  4 

a— 5     /a?  —  4     a?  —  12      ic-f2" 


18 


/a; -4  _  a;-  32  _  x  -f  2\_  Q 
^20  3  24    /      ' 


30.   8~|(10-.*)+|(l5-«)»|(18-*)-S^lS. 

o  o  o  a? 


315.  Literal  Equations.  An  equation  in  which  some  of  the 
numbers  that  are  regarded  as  known  numbers  are  expressed  in 
literal  notation  is  a  literal  equation. 

1.    Solve  «_1=J_9. 


Solution.  —  1  = 9. 

x  x 

L.C.  D.  =x. 
a  —  x  —  h  —  9x. 

Sx  =  b  —  a. 

X~     8 


202  Equations  Containing  Fractions 

2.   Solve      ?-i(9a-3a0-^±*  =  i2^«. 

a     3V  y        2a  a 

Solution.  *-  3a  +  *-"-+*  =  4a  ~  x. 

a  2a  a 

L.  C.  D.  =  2  a. 

2z  -  6  a2  +  2  asc  -  (a  +  *)  =  2(4  a  -  as). 

2a-6a2  +  2oz-a-x  =  8a-2x. 

3x  +  2ax  =  6a2  +9a. 

(2  a  +  3)x  =  3  o(2  a  +  3). 

x=3a. 

EXERCISE 

316.    #oZ?;e  £fte  following  literal  equations,  regarding  x  as  the 
unknown  number  in  each : 


,     a      6 

1. =  c 

X        X 

5. 

x  —  a              #  —  6 

m  = n 

a                     b 

2.    -  —  -  =  a 

a      6 

6. 

a      b  +  x  =  b      a  +  aJ. 
b                    a 

3.   »--  =  &. 

a 

7. 

-  —  b  =  -  —  a. 
a            6 

.     a  —  bx  .   , 

4. h  c 

c 

6c 

c 

# 

8. 

a+6_c=d      a~6 
05                                o; 

9     * 

-a 

2 

»-». 

x  — 

c-0 

6c      ' 

1 

ac 

ab 

10.    » 

c 

+ 

x  4-  ac 
6 

+x 

+  6c  =  o. 

a 

11.   a(m-?)=6(«-g. 

10     a—  bx  .b  —  ex  .c  —  ax_Ci 

14. 1 1 —  —  V. 

be  ac  ab 

ax  -\-b  ^  d__b      ex  -f  d 

x  a      a  x 

14-  t^x+1)-k(5a*-4b)=t- 


Equations  Containing  Fractions  203 

a  —  bm      c  —  bn  _  -. 
mx  nx 

16.  (---x\(a  +  x)  —  (-  +  x\a  —  x)=0. 

17.  *±±  +  h-^±=(a  +  *)+(&-  a>). 

3  6(a?  —  a)  ,  x  —  b2      6(4  a  +  ca;)  _  q 
5  a  15  &  6  a 


^tHj.^-^.^ 


a« 
&         2c  3d 


20     a(ft  —  a?)  ,  &(c  —  a)  =  a  +  &  _  fb     a 
bx  ex  x         \c      b 

In  equations  21  to  34  solve  for  each  letter  involved  in  terms  of 
the  others. 

21.  i-i=J. 

D     d     f 
Solution.  L.  C.  D.  =  Ddf. 

df-Df=Dd. 
Solving  for  /,  (d  -  D)f  =  Dd, 

.    f=    Dd 
"J      d-D 
Solving  for  2>,  (-  d  -  f)D  =  -  df, 


.  •.  D  = 


df 


d+f 
Solving  f or  d,  (f-D)d  =  Df, 

ovd=  -M. 


f-D 

22.  3x-5y  +  7z  =  %(x-y  +  3z). 

23.  a-4  =  (p4-3a)(a  +  2). 

24.  s=?(a+i  27-  y=mx  +  c- 

2  28.   lx+  my  =  1. 

25.  Z  =  a+(7i-l)d.  29.   ^  +  £?/  +  (7  =  0. 

26.  S  +  f.l  30.    i-i-li 

a     6  10     »     a; 


204  Equations  Containing  Fractions 

Solve  the  following  equations  for  each  letter  involved : 

31.  A -<&+$.  33.    *=*b«. 

z  a  —  b 

32.  c=^*.  34.    T  =  ±+t. 

b  —  a  a 

317.  If  an  equation  contains  fractions  with  polynomial  de- 
nominators, find  the  L.  C.  D.,  and  proceed  as  in  the  preceding 
problems. 

3  12 

1.   Solve  the  equation 1- 


9      a? -|-3     3-x 

Arrange  the  denominators  in  descending  powers  of  x  and  factor  them 
to  find  the  L.  C.  D. 

Solution.  3      + 


x2-9     a;  +  3     x  —  3 
L.  C.  D.  =  (>  +  3)(x-3). 
Multiply  every  term  of  both  members  of  the  equation  by  the  L.  C.  D. 
to  clear  of  fractions. 

3 +(a5-3)  =-  2(s  +  3). 
3+z-3=-2x-6. 
z  +  2x=-3  +  3-6. 
3x=-6. 
x=-2. 

Check.  — ^-  + 1 =  -*— ,  or  ?  =  ?. 

4-9-2+3     3  +  2'        5     6 


2.    Solve 
Solution. 


l-2s     5-6z  =  8(l-3z2) 

3  _  4 z     7-Sz     3(21  -  52  z  +  32 z2) 


l_2g      5-6g_         8(1-  3  gg) 
3_4g     7-8*     3(3-4g)(7-8g)' 
The  L.  C.  D.  =  3(3  -  4  g)(7  -  8  «>. 
3(1  -  2  g)(7  -  8  g)  -  3(5  -  6  g)(3  -  4  g)  =  8(1  -  3  g2).  (Why  ?) 

21  _  66  g  +  48  g2  -  46  +  114  g  -  72  g2  =  8  -  24  #, 


Equations  Containing  Fractions  205 

-  66  z  +  114  z  -  8  -  21  +  46. 
48  z  =  32. 
»  =  f 

Check.        !=i_$rJ  = »(  *~  t) or  -§=-§. 

3-f     7-V-     3(21 -■4A  +  H*)  5         ^ 

EXERCISE 
318.   /Sofae  the  following  equations: 


1. 

1              7      _2             £       16             13         .- 

x  +  2     3x  +  6     3                ' l-3a;     l-3a; 

2. 

9               7       _13                 5              3      _5 

2a;  +  2     3a?  +  3      12           '  a?  +  l     2  x  +  2     2 

3. 

7               11                         9*  +  7      /       «-2\_3 

8a;  +  2     20x  +  5~13'      6'        2           \j"         7    J     * 

7. 

a;  +  3     x—2     3x-5     1 
2             3            12     +4 

8. 

60-x     5x-5_6     24-3a; 
14              7                       4 

9. 

0        -,      7a;  — 2     3a;  +  4      Ix  —  4     5a;+l 
3               5               5-8 

10. 

3         a?  +  l         a?             „     6a;+7      2a;-2     2a;+l 

x  +  1     x  —  1     1  —  x2  15         7  a;— 6 

Hint.  If  some  of  the  denominators  are  monomials,  it  is  best  to  clear 
the  equation  of  the  monomial  denominators  first  and  then  collect  terms 
before  clearing  the  equation  of  the  polynomial  denominators.  In  exer- 
cise 11  proceed  as  follows  : 

6  x  +  7  -  15(2*-2)  _  6  x  +  3      (Multiplying  by  15.) 

7x  —  6 

4  -  30  x  —  30  =  0.     (Collecting  terms  after  transposing.) 

4(7  x  -  6)  -  (30  x  -  30)  =  0.     (Multiplying  by  7  x  -  6.) 
28  x  -  24  -  30  x  +  30  =  0. 
-  2  a;  =  -  6. 
a;  =  3. 


206  Equations  Containing  Fractions 

Solve  the  following  equations  : 

12     55      79-2a;^a;  +  3  5       10-7  a;  =  13  +  15  a; 

3x     60-2a;         a;  '3a;      6-7  a;  15a; 

14    8a;+5      3-  7  a;  =  16a;  +  15      2\ 

14         6  a;  +  2  28  7  " 


15. 


16. 


8  a; +  37      7  a;  -  29  =  4a;  +  12 
18  5a;-12~~        9 

y  —  5  _  y  +  5  _         21  y 
2/  +  5      y-h~     25-y2' 


1  ly  O  .  O  X  « 

2-a?     2  +  x~  x2-±~ 


18. 


5    ^L 3        1     =  1  _      1 

4  "  a;  +  4     4  '  a  +  2     2  *  a;  +  6 


19.  2x_3^-3  =  3_l-i^> 
5  a;  + 1  10 

20.  _L_         3     +     I8— 0. 
a;  - 1      1  +  a;      1  -  a;2 


-H) 


5  V       5/  a; +  2      a;     a;(a;  +  2) 

7a;+26     17  +  4a;  =  10-a;      13  +  a; 
'     a; +  21  21  3  7 

1-a;     3-a?=  6a;  +  5       l+8a; 
'       3     +     5         8  a? -15         15 

1_2a?  +  l        a-11   =0 
3a;- 15     2a;- 10 

3a;-5     5a;-l      a^-4=2 
5*-5    f*-7     a?-l 

29.    -**-:=  2- f      7a?      +       »     V 
6x  +  2  \15a;  +  5     3a;  +  V 


Equations  Containing  Fractions  207 

30. 


31. 
32. 
33. 


37. 
38. 


40. 
41. 


2    4a;+l      1     2a;-l       3a?  +  2=0 

x-2       3  '   x-2      5»-10 
4 -2  a;  4  3  4«2 


3  6x-S     2x-\      6a;-3 

3  4a?-5  =  5     7a-3 

4  '  3a;-7     7  '  5a;-4* 

*±4(3a?-ll)=3(aj-3). 
a?—  1 

34.  6(x  -  6)=  3  g  ~  14(2  a?  -  11). 

a;  —  4 

35.  3^1  +  3^+1     _3^-67_  =  5# 
3a;  +  l       »-l       3a;2-2a;-l 

36.  2^-3     3a;  +  5  a;2 -11      _  * 


a;_l         aj-2       a;2 -3  a; +  2 
2  a;  2  a;2  +  7  2 


x-2      a;2  -  3  a;  +  2      a?  - 1 

4  a?  +  5     2(a?  -  2)  _      -  7 
2  a; +  6        a; -3        6  a;2 -54 


39.    -5 3_       2a;  +  2    _1  =  Q 

Sx-6     2»-4     3a?-6»     3 
2  a;  + 1        2  a;  -  1  _      9  a;  +  17 
2  a; -16     2  a; +  12     x2  -2  a?  -48* 
a;  —  2q       a?4-2&_3a?  —  3  a 
a  +  6      2a  +  26~~       26 

42  a*  —  4  5a;  ,  52  —  aa;      o 

a2  +  4  6       62  +  a 

43  2a;  +  a     3a;2-22q2=5 
a?  +  3  a        a;2 -9  a2 

44  2  a  —  a;  _  5  +  x  _5a  -±  x  _x  +  6 

a-5  3       ~ a+2  2 

._         1      .  a  +  &         1      .a  —  6 

45.    — —  -\ = -H 

a  +  o         a;         a  —  o         x 


208  Equations  Containing  Fractions 

Solve  the  following  equations  : 

46.    ^=^  +  ^1^4-2  =  0. 
b  a 


47. 
48. 


a  b  62 


b      &-bx 


4.50-26)^3       b2-5bx 
Sx-3b        2     6x2-6bx 


PROBLEMS  LEADING  TO  FRACTIONAL  EQUATIONS 

319.  1.  What  number  added  to  both  terms  of  the  fraction 
-|  will  give  a  fraction  whose  value  is  -|  ? 

Solution.  Let  x  =  the  required  number. 

Then   |il?  =  -  •     ( By  the  conditions. ) 

5  +  x     9       v   J  J 

.-.  18+9x  =  40+8z. 

.*.  x  =  22,  the  required  number. 

Check.  2  +  22  =  24  =  8 

5  +  22     27      9 

2.  The  numerator  of  a  fraction  exceeds  the  denominator 
by  20,  and  if  7  is  added  to  both  terms  of  the  fraction,  the  value 
of  the  resulting  fraction  is  3.     Find  the  original  fraction. 

3.  What  number  added  to  both  terms  of  the  fraction  f  will 
double  the  value  of  the  fraction  ? 

4.  The  sum  of  the  numerator  and  the  denominator  of  a 
fraction  is  20.  If  the  numerator  is  multiplied  by  2  and  the 
denominator  diminished  by  3,  the  resulting  fraction  is  equal  to 
-J-.     What  is  the  original  fraction  ? 

5.  The  difference  between  two  numbers  is  16,  and  the 
quotient  of  the  larger  divided  by  the  smaller  is  2\.  What  are 
the  numbers  ? 

6.  |  of  what  number  exceeds  -|  of  the  same  number  by  1  ? 


Equations  Containing  Fractions  209 

7.  In  a  division  the  dividend  exceeded  the  divisor  by  52, 
the  quotient  was  6,  and  the  remainder  was  8.  Find  the  divi- 
dend and  the  divisor. 

8.  Divide  72  into  two  parts  such  that  f  of  one  part  shall 
exceed  \  the  other  part  by  26. 

9.  A  man  made  a  journey  of  40  miles  in  4|  hours.  Part  of 
the  way  he  traveled  in  an  automobile  at  20  miles  an  hour  and 
the  remaining  distance  he  walked  at  the  rate  of  4  miles  an 
hour.     How  far  did  he  ride  ? 

Solution.  Let  x  =  number  of  miles  he  rode. 

Hence  40  —  x  =  number  of  miles  he  walked. 

Also  —  =  number  of  hours  he  rode, 
20 

and  — ^—  =  number  of  hours  he  walked. 
4 

Then  —  +  40  ~  x  =  4*.     (By  the  conditions.) 
20  4  s      v    j  / 

Solve  the  equation. 

10.  A  vessel  that  ordinarily  goes  16  miles  an  hour  is 
obliged  to  slacken  to  half  speed  during  a  part  of  a  trip  of  130 
miles,  thereby  requiring  10  hours  to  make  the  trip.  For  how 
long  a  distance  was  it  traveling  under  reduced  speed  ? 

11.  If  one  man  can  do  a  piece  of  work  in  8  days  and  an- 
other man  can  do  the  same  work  in  6  days,  how  long  will  it 
take  both  men  working  together  ? 

Solution.     Let  x  =  number  of  days  for  both. 

Hence  -  =  the  part  of  the  work  both  can  do  in  one  day. 

Theni  +  Ll. 

8      0     x 

Let  the  student  explain  the  equation  and  solve  it. 

12.  A  can  do  a  piece  of  work  in  5  days ;  B  works  only  half 
as  fast  as  A.     How  long  will  it  take  both  working  together  ? 


210  Equations  Containing  Fractions 

13  A  can  do  a  piece  of  work  in  12  days,  but  with  B's  help 
he  can  do  it  in  8  days.  How  long  would  it  take  B  if  he 
worked  alone  ? 

14.  A  tank  has  two  inlet  pipes.  One  can  fill  it  in  40 
minutes  and  the  other  in  60  minutes.  How  long  will  it  take 
if  both  are  running  at  the  same  time  ? 

15.  A  tank  has  two  inlet  pipes  numbered  1  and  2,  and  two 
discharge  pipes,  3  and  4,  with  the  following  capacities  :  1  run- 
ning alone  can  fill  the  tank  in  60  minutes ;  2  alone  can  fill  it 
in  80  minutes ;  3  alone  can  empty  it  in  72  minutes,  and  4  can 
empty  it  in  40  minutes. 

(a)  Beginning  with  the  tank  empty,  how  long  will  it  take 
land  2  to  fill  it? 

(6)  Beginning  with  the  tank  full,  how  long  will  it  take  3 
and  4  to  empty  it  ? 

(c)  Beginning  with  the  tank  full  and  all  pipes  flowing,  how 
long  will  it  take  to  empty  it  ? 

(cZ)  Beginning  with  the  tank  empty,  how  long  will  it  take  to 
fill  it  if  1,  2,  and  3  are  flowing  ? 

(e)  Beginning  with  the  tank  half  full,  will  it  be  filled  or 
emptied,  and  after  how  long,  if  2,  3,  and  4  are  flowing  ? 

16.  What  amount  of  money  drawing  simple  interest  at  5  % 
will  amount  to  $  287.50  in  3  years  ? 

Solution.     Let  x  =  number  of  dollars  on  interest. 

Hence  —  =  number  of  dollars  of  interest  per  year, 
100 

and  i— ^  =  number  of  dollars  of  interest  in  3  years. 
100 

Then  x  +  —  =  287.50. 
100 

.  .  100  a;  +  15  a;  =  28750, 

or  115  a;  =  28750. 

.-.  x  =  250. 

Therefore  the  original  principal  was  $  250. 


Equations  Containing  Fractions  211 

17.  What  was  the  face  of  a  note  drawing  4  %  simple  interest  if 
it  took  $132.50  to  settle  the  note  18  months  after  it  was  given? 

18.  A  man  loaned  $  800  in  two  parts,  one  part  yielding  5  %  per 
annum  and  the  other  part  yielding  6  % .  The  interest  amounted  to 
%  44.50  per  year  on  the  two  notes.    How  was  the  money  divided  ? 

19.  A  man  received  $  665  for  an  automobile,  which  was 
30  %  below  its  original  cost.     How  much  did  it  cost  ? 

20.  How  much  water  must  be  added  to  80  pounds  of  a  5 
per  cent  salt  solution  to  obtain  a  4  per  cent  solution  ? 

Solution.  Evidently  it  will  require  the  addition  of  water  to  change 
the  solution  from  5  per  cent  salt  to  4  per  cent  salt.  The  amount  of  the 
salt  is,  therefore,  the  same  in  both  solutions,  and  we  may  use  this  fact  as 
the  basis  of  an  equation. 

Let  x  a  number  of  pounds  of  water  to  be  added. 
Hence  80  -f  x  =  number  of  pounds  of  salt  and  water  in  the  new 
solution, 
and  yj^  (80  -f  x)  =  number  of  pounds  of  salt  in  the  new  solution. 
Also  jf^  •  80  =  number  of  pounds  of  salt  in  first  solution. 
Then  T$5  (80  +  x)  =  T^  •  80.     (Since  there  was  the  same  amount  of  salt 

in  both  solutions.) 
.*.  4(80  +  x)=  5  •  80,     (Clearing  the  last  equation  of  fractions.) 
or  320  +  4  x  =  400. 
.-.  ix  =  80. 
.  •.   x  =  20,  the  number  of  pounds  of  water  required. 

21.  How  much  salt  must  be  added  to  80  pounds  of  a  5  % 
salt  solution  to  change  it  to  a  10  %  salt  solution  ? 

Solution.     Let  x  =  number  of  pounds  of  salt  added. 

Hence  80  +  x  =  number  of  pounds  of  salt  and  water  in  new  solution, 
and  t$j(80  +  x)  =  number  of  pounds  of  water  in  new  solution. 
Also  ^  •  80  =  number  of  pounds  of  water  in  original  solution. 
Then  ^(80  +  *)=^- 80. 
.-.  90(80  +  x)  =  95-  80, 
or  7200  +  90  x  =  7600. 
.-.  90  a;  =  400. 

.-.  x  =  4$,  the  number  of  pounds  of  salt  required. 
Query.     Why  is  it  not  sufficient  merely  to  double  the  amount  of  salt  in 
order  to  double  the  strength  of  the  solution  ? 


212  Equations  Containing  Fractions 

22.  How  much  salt  must  be  added  to  100  pounds  of  a  10  % 
salt  solution  to  change  it  to  a  12  per  cent  solution  ? 

23.  How  much  water  must  be  added  to  change  100  pounds 
of  10  %  salt  solution  to  a  4  %  salt  solution  ? 

24.  How  much  water  must  be  added  to  each  ounce  of  a 
90  °/0  alcohol  solution  to  reduce  it  to  a  60  %  solution  ? 

25.  A  merchant  marked  an  article  $  8  and  gave  20  %  dis- 
count. Another  merchant  marked  the  same  article  at  a  higher 
price  but  gave  33J  %  discount.  Find  the  marking  price  of  the 
second  merchant,  if  the  discounted  price  was  the  same  for  both. 

26.  If  100  pounds  of  sea  water  contain  2.6  pounds  of  salt, 
how  much  fresh  water  must  be  added  to  make  a  new  solution 
30  pounds  of  which  shall  contain  .6  of  a  pound  of  salt  ? 

27.  The  sum  of  two  numbers  is  70.  If  14  is  subtracted 
from  one  of  them  and  added  to  the  other,  the  quotient  of  the 
numbers  is  inverted.     What  are  the  numbers  ? 

28.  The  population  of  a  city  increased  each  year  5  %  of  the 
population  of  the  preceding  year.  It  now  has  194,481  inhabit- 
ants.    What  was  the  population  3  years  ago  ? 

29.  Find  two  numbers  whose  sum  is  s  and  whose  quotient 

is*. 

b 

30.  Divide  the  number  144  into  two  parts  such  that  one  part 
shall  be  §  of  the  other. 

31.  The  numerator  of  a  fraction  is  35  less  than  its  denomina- 
tor. If  both  the  numerator  and  the  denominator  are  increased 
by  2,  the  fraction  is  equal  to  f .     Find  the  fraction. 

32.  The  cost  per  ounce  of  gold  in  December  1914  was  about 
41  times  that  of  silver.  Find  the  cost  per  ounce  of  each  if 
8.5  oz.  of  silver  and  \  oz.  of  gold  together  cost  $  14.50. 

33.  A  watch  chain  weighing  }  oz.  is  made  of  platinum  and 
gold.     How  much  of  each  metal  is  in  the  chain  if  the  gold  is 


Equations  Containing  Fractions  213 

worth  $  20  an  ounce  and  the  platinum  is  worth  $  48  an  ounce 
and  the  total  value  of  the  metal  in  the  chain  is  $  22.75  ? 

34.  A  man  invests  $4500,  part  at  6  %  and  part  at  5%. 
The  total  income  from  the  two  investments  is  $  245.  Find  the 
amount  invested  at  each  rate. 

35.  A  certain  sum  of  money  is  invested  in  a  6  %  mortgage 
and  $  500  more  than  this  sum  is  invested  in  4  %  bonds.  If  the 
incomes  from  the  two  investments  are  the  same,  how  much  is 
invested  in  each  ? 

36.  An  estate  of  $  12,000  is  divided  among  three  heirs. 
The  first  receives  §  as  much  as  the  second  and  the  third 
receives  $  400  more  than  the  second.  How  much  does  each 
get? 

37.  A  man  can  paint  a  house  in  6  days ;  his  son  can  paint 
it  in  16  days.  How  many  days  would  it  take  both  working 
together  ? 

38.  A  football  team  wins  a  game  by  14  points  and  the 
losing  team  scores  4  less  than  half  as  many  points  as  the 
winning  team.     What  is  the  score  ? 

39.  The   pressure  of  water  at  a  depth  of  d  feet  on  each 

square  inch  is  given  in  pounds  by  the  formula  P  (pressure) 

62  5 
=  — -{—  d.     If  the  pressure  of  the  air  at  the  surface  is  14  pounds 

per  square  inch,  at  what  depth  will  it  be  10  times  as  great  ? 

40.  It  is  1024  miles  from  Chicago  to  Denver.  A  train  that 
usually  averages  32  miles  an  hour  is  delayed  2  hours  by  an 
accident,  but  by  running  12  miles  an  hour  faster  just  makes 
up  the  lost  time.     How  far  did  it  run  at  each  rate  ? 

41.  A  dairyman  wishes  to  mix  milk  containing  5  %  butter 
fat  with  cream  containing  30  %  butter  fat  to  get  a  mixture 
containing  20%  butter  fat.  How  much  of  each  should  be 
taken  to  get  10  quarts  of  the  mixture? 


214  Equations  Containing  Fractions 

42.  Any  volume  of  aluminum  weighs  -f-  as  much  as  the 
same  volume  of  cast  iron.  When  i  of  the  cast  iron  of  a 
gasoline  engine  is  replaced  by  aluminum  parts  of  the  same  size, 
the  weight  of  the  engine  is  320  pounds.  What  was  the 
original  weight  of  cast  iron  ? 


REVIEW   OF    FRACTIONS    AND    FRACTIONAL    EQUATIONS 

320.    1.   What  is  the  rule  for  adding  fractions  ? 

2.  How  do  we  "  clear  an  equation  of  fractions  "  ? 

3.  What  principle  is  involved  in  "  clearing  an  equation  of 
fractions  "  ? 

4.  How  is  a  fraction  multiplied  by  an  integer  ? 

6.  Simplify  -li+-l-+-l_. 

Note.     The  student  should  note  that  example  6  is  not  an  equation, 
and  that  he  is  not  to  clear  fractions.  \ 

7.  Why  do  you  have  trouble  if  you  try  to  solve  the  equation 

1         l  =  2(s  +  l)9 

05  +  2        X        05(8  +  2) 

8.  Solve  (5+g(5-Q  +  |=05  +  12. 

9.  Simplify(5+|J+(5-|J+|. 

10.  Solve  -2— L± i-=0. 

05  —  9        05  —  5        05  —  4 

11.  Simplify—?- ?L  + 


05  —  1        05  —  2        05  —  3 

a     b     c      b 


12.   Solve 

05       c      05      a 


Review  of  Fractions  215 

13.  Solve  for  t,  ^=&c  +  -- 

z  z 

14.  What  number  must  be  added  to  f  to  get  the  same  result 
that  would  be  obtained  by  multiplying  it  by  %  ? 

15.  -*-+-* §-=0. 

x-YI     x-19     jb-18 

1C     flJ-8,a;-3  .  oj-9     a;-  1  .  a?-  13  .  a;-  6 
#—3     x—5     x—7     x  —  3      x—5      x—7 

Note.     First  transpose  the  fractions  and  combine  each  pair  having  the 
same  denominator. 

17     x  +  2  .  x  +  1     x  +  l=x-\-9     x-3     x  +  4= 
x  +  7     x  +  5     x+3     x+1     x  +  5     x  +  3 


18. 


19. 


q2-62  .  q2  +  2qft  +  &2 
a3  _  53  *    a2  +  ab  +  52  * 

3?  -  7  a;  +  12  ^    x*-l    .  a^  +  a?  +  l. 
x2  —  x  x2  —  4  a?  a,*2 


20.    Reduce  to  lowest  terms  S^  +  ^  +  s 

a?-l 
a4  —  a3  —  a  +  1 


21.   Reduce 


q4  +  a3  -  q  -  1 


22     5  +  3a:     5-3a;     48-2a; 
2-x        2+a;        a?-4 

23.  Solve  x-x-^  =  5§-x  +  10  +  g. 

3  5  4 

24.  Simplify 


12  a;2  -f-  £  -20     12  a;2 -h  25  a;  +  12 

25.  Solve  1.2  a;  -  .05  =  .07  x  +  .3x+  16.55. 

a  c 

26.  6  +  C       -     a~*~& 


1  -j-  k  +  c      1  1  c  +  a 
a  4-  6  6  +  c 


216  Equations  Containing  Fractions 

27.    Prove  that  (*±1  -  1  Vf — _  -ii 

V^-i      y    Vs-1    2- 


a; 


s-2 


x(x-\-l)(2x-S) 


28.  Show  that  ^+y^/2^3^4g-9|!\ 

i^-a,    [x^f     tf-f )    *  +w 

q  12         10 

29.  Show  that  x  =  —  satisfies  the  equation l —  =  64-. 

19  u  a;      it;  —  1        3 


3°*    U2+1+2/2A2/     *J 


2-1=* 


5 


31.  ■      — gi =  1.     Solve. 

32.  Snnphfy  t  •  C-3 


1-JI 


1-1 


(Princeton.) 
33.    Simplify  (<*  +  1  +  ^W»*±*?  -  1 V 

.    (Sheffield  Scientific  School.) 
a2  +  b2 


b  a?-b2 


34.   Simplify  _      -  .  ^     (Yale.) 
_c      ajj   '|_c2      a2_|' 


35.    Simplify 


6 

a6  4-  c6 
a3c3 


Verify   the   result   by  using  a  =  2,   c  =  1    in   the   original 
fraction  and  in  the  answer.  (Yale.) 

»■  "-»Hs5I+i{:-3-,^5*)} 

(Princeton.) 


Review  of  Fractions  217 

37.  Simplify(a2+-^)(a2  +  .^(^  +  ^_). 

(Sheffield  Scientific  School.) 

38.  Solve  2(*-a)+8(*-»)  =  6. 

b  a 

39.  Simplify  x  +  1  +  — g*        5s-4t 

„n    a  i        7,1      23  -6,7        1 

40.  Solve  _  +  _  =  __  +  _-_. 

41.  Show  that  (100a  +  10y  +  z)^3  =  33a  +  3y  +  a;  +  3f  +  g, 

o 

and  from  this  equation  show  that  if  the  sum  of  the  digits  of  a 
number  of  three  figures  is  divisible  by  3,  the  number  itself  is 
divisible  by  3.  Show  in  the  same  way  that  any  number  of 
four  figures  is  divisible  by  3  if  the  sum  of  its  digits  is  divisible 
by  3. 

42.  Show  similarly  that  if  the  sum  of  the  digits  of  a 
number  is  divisible  by  9,  the  number  itself  is  divisible  by  9. 

43.  Any  number  ending  in  5  can  be  written  as  10  a  +  5, 
where  a  is  the  tens'  figure.  (10  a  +  5)2=  100a2  +  100  a  +  25  = 
100  •  a(a  +  1)  +  25.  From  this  we  may  get  the  squares  of 
numbers  of  two  figures  ending  in  5  by  multiplying  the  first 
figure  by  1  more  than  itself  and  writing  the  product  before  25. 
Thus,  652  =  4225.  (6x7  =  42.)  Square  all  numbers  of  two 
figures  that  end  in  5. 

44.  (a  4- 1)2  =  a2  +  2  a  +  1.  The  square  of  a  -f  1  exceeds 
the  square  of  a  by  2  a  +  1.  This  means  that  the  square  of 
21,  20  +  1,  exceeds  the  square  of  20  by  2  •  20  +  1,  and  therefore 
212  =  441.     Square  31,  41,  51,  61,  etc. 


XIL  RATIO  AND  PROPORTION 

321.  Ratio.  The  quotient  of  one  number  divided  by  another 
number  of  the  same  kind  is  their  ratio.  The  former  number  is 
the  antecedent  and  the  latter  is  the  consequent. 

The  ratio  is  usually  written  in  the  form  of  a  fraction  and  its 
terms  bear  the  same  relation  to  each  other  as  the  numerator 
and  the  denominator  of  a  fraction. 

$  10 
Thus,  - —    represents  the  ratio  of  $10  to  $5.     The  value  of  this  ratio 

$5      • 

is  ^,  or  2.   -  represents  the  ratio  of  a  to  b.     It  is  usually  read,  the  ratio 
b 

of  a  to  6  or  a  divided  by  b.     The  above  ratios  are  also  sometimes  written 

$  10  : 1 5,  and  a  :b.    The  colon  is  used  here  as  a  sign  of  division. 

The  value  of  a  ratio  is  always  an  abstract  number.     (Why  ?) 


ORAL  EXERCISE 

322.    Bead  the 

following  ratioi 

\  an 

d  give 

their  values  : 

.     $6 
'    $8* 

5.    7  men  : 

21 

men. 

q    ma 
na 

2     2a2. 
Sab 

6     — . 
my 

10.   21:. 75. 

3     $15 

$6 

7       1.1 
■•      4-2"- 

11.    ,3Aft-  • 
10  in. 

4   ii§. 

8.    X-:  y- 

y   x 

12.    2  yd.  :  2  ft, 

13.   If  the  ratio  of  x  to  3  is  equal  to  5,  what  is  the  value  of  x  ? 


Hint.    ^  =  5.     Solve. 


3 

14.   If  the  ratio  of  a;  to  -J-  is  equal  to  2,  what  is  the  value 
of  a? 

218 


Ratio  and  Proportion  219 

15.  What  number  bears  to  5  the  ratio  .3  ?     I  Solve  ^  =  .3.  j 

16.  Can  you  express  a  ratio  between  $12  and  4  f t.  ?  4  bu. 
and  2  qt.  ?    1  rd.  and  1  in.  ?    10  sq.  in.  and  2  cu.  in.  ? 

Simplify  the  following  ratios  by  treating  them  as  fractions  and 
reducing  them  to  their  lowest  terms : 

17.  (ra2  —  n2)  :  (m  +  n).  18.    x3  —  y3  :  x  —  y. 
19.   Which  ratio  is  the  greater,  -f-  or  -J  ?   f  or  ^  ? 

323.  Proportion.     An  equality  of  two  ratios  is  a  proportion. 

Thus,  ^  =  f-f  is  a  proportion.     Also  -  =  -  is  a  proportion,  if  a  and  6 

b      d 

are  the  same  kind  of  numbers,  and  c  and  d  are  also  the  same  kind  of 

numbers.     This  proportion  is  read,  the  ratio  of  a  to  b  equals  the  ratio  of  c 

to  d.     The  proportion  is  also  sometimes  written  a  :  b  =  c  :  d,  or  a  :  6  :  :  c  :  d. 

These  proportions  may  be  read,  a  is  to  b  as  c  is  to  d.     The  fractional 

form  is,  however,  much  more  commonly  used. 

EXERCISE 

324.  1.   What  value  must  be  given  to  d,  if  a  =  1,  6  =  2, 

c  =  3,  in  the  proportion  -  =  -? 

b     d 

2.  What  is  the  value  of  d  if  a  =  2,  6  =  3,  c  =  4  ? 

3.  a2-62:a-6  =  ? 

4.  Divide  60  into  two  parts  that  are  in  the  ratio  of  2  to  3. 
Hint.     Let  x  and  60  —  x  be  the  two  numbers. 

325.  Terms  of  a  proportion.  The  four  numbers,  a,  b,  c,  and 
d  are  the  terms  of  the  proportion  a :  b  =  c :  d.  The  first  and 
fourth  terms,  a  and  d,  are  the  extremes,  and  the  second  and 
third  terms,  b  and  c,  are  the  means.  The  first  and  third  terms, 
a  and  c,  are  the  antecedents,  and  the  second  and  fourth  terms, 
b  and  c?,.  are  the  consequents. 


220  Ratio  and  Proportion 

x         2 
In  the  proportion =  -  ,  name  the  extremes,  the  means, 

X  —  4       o 

the  antecedents,  the  consequents. 

326.  Fourth  Proportional,  Third  Proportional,  and  Mean  Pro- 

ft  Q 

portional.     The  fourth  term,  d,  of  the  proportion  -  =  -  is  the 

fourth  proportional  to  the  other  three  terms  taken  in  the  order 

a,  b,c. 

In  the  proportion  -  =  - ,  where  the  means  are  equal,  c  is  a 
b      c 

third  proportional  to   a  and  b,  and  b  is  the  mean  proportional 

between  a  and  c. 

ORAL  EXERCISE 

327.  In  the  following  proportions  name  the  extremes,  the 
means,  the  antecedents,  the  consequents,  the  fourth  proportionals, 
the  mean  proportionals,  and  the  third  proportionals. 

.     2      4  .     a      b 

1.  —  SB-.  4.      -  =  -• 

3      6  be 

2  S 

2.  -  =  — - .  5.    m  :  p  =  q  :  s. 

3  4.5  F 

3.  a:b=  c:d.  6.    x  :  y  =  y:  z. 

328.  A  proportion  may  be  treated  as  an  ordinary  frac- 
tional equation.  The  unknown  number  may  be  in  any  term  of 
the  proportion. 

3     5 

Solve  the  proportion  -  =  -  for  x. 
i      x 

Solution.  -  =  -. 

7     x 

3x  =  35. 

z  =  llf 

Check.     Substitute  llf  for  x  in  the  proportion. 


Ratio  and  Proportion  221 

EXERCISE 

Solve  for  x  in  each  proportion  : 

.       x    3  =  12  51  =  68. 

a;     16  15       x 

3.    6.3  :  x  =  13|- :  20.     (Write  in  fractional  form.) 

4     ??=._£  5      8^=   bc 

95     57*  '       a;        \ac 

6.  Find  the  fourth  proportional  to  (a)  3,  4, 6:  (-  =  -");  (&)  2, 
4|,9i;  (c)  a,b,c.  V4      ^ 

7.  Find  the  third  proportional  to  (a)  9  and  6 ;  (b)  a2  —  b2 
and  a  —  b  ;  (c)  a  and  b. 

8.  Divide  120  into  two  parts  which  are  in  the  ratio  of  2  to  3. 

Hint.     Let  x  and  120  —  x  represent  the  two  parts.     Why  ? 

/» 

9.  Divide  182  into  two  parts  whose  ratio  equals  -• 

7 

10.  What  number  added  to  both  terms  of  the  ratio  -  will 

2  8 

give  a  ratio  whose  value  is  -  ? 

o 

11.  Find  a  mean  proportional  between  2  and  8. 

2      x 
Solution.     The  equation  is  -  = — . 
x     8 

z2  =  16. 

x2-  16=0,  or  (z-4)(x+  4)=0.  (§239.) 

x  =  4  or  —  4. 

12.  Find  a  mean  proportional  between  : 

(a)  2  and  18.  (c)  —  and  -• 

sc  a 

(b)  3  and  27.  (d)  ^L±_^.2  and  p  -  q. 

13.  Divide  $  180  between  two  men  so  that  their  shares  will 
be  in  the  ratio  of  13  to  5. 

Hint.     See  example  8,  or  let  13  x  and  5  x  represent  the  two  shares. 


222  Ratio  and  Proportion 

14.  Divide  $  180  among  three  men  so  that  their  shares  shall 
bear  to  each  other  the  relation  2:3:5. 

Hint.  This  notation  means  that  the  first  man's  share  is  to  the  second 
man's  share  as  2  is  to  3.  Also  the  first  man's  share  is  to  the  third  man's' 
share  as  2  is  to  5.    The  shares  may  be  represented  by  2x,  3  a:,  and  5x. 

15.  Solve  for  a?,  ^=4  =  f- 

'  x  +  3     6 

16.  Solve  for  y,    y  —  7  :  y  —  3  =  y  —  11  :  y  —  9. 

A  C     B         X  17-    In  the  figure  .40  =  9  inches, 

1 1 ' 1        CB  =  3     inches,     and      BX=x. 

Find  a  if  AC :  CB  =  AX :  BX. 


PROPERTIES  OF  PROPORTIONS 

2  8 

330.    Consider  the  proportion  -  =  — .     Cleared  of  fractions 

3  1— 

this  gives  2  •  12  =  3  •  8.     This  illustrates  the  following  impor- 
tant property  of  any  proportion  : 

I.  If  four  numbers  are  in  proportion,  the  product  of  the  means  is  equal 
to  the  product  of  the  extremes. 

Proof.     Let  a,  b,  c,  and  d  be  four  numbers  in  proportion. 

Then  2  =  1- 
b     d 

.-.  a  •  d  =  b  •  c.     (Clearing  of  fractions.) 

The  last  equation  states  that  the  product  of  the  means  in 
any  proportion  equals  the  product  of  the  extremes.  This  is  a 
test  of  the  correctness  of  a  proportion,  or  of  the  equality  of 
two  ratios. 


Find  the  value  of  x  in  : 

1.    2:  #  =  3:6. 

2. 

x  :  4  =  3  :  6. 

3.    a  :  b  =  c  :x 

3  x  =  12. 

6x=  12. 

ax  =  be. 

»  =  4. 

x  =  2. 

•  «!£. 

a 

7. 

- :  x  =  b  :  ab. 
a 

8. 

ab*  a2b              1 
c     5c*           106c 

9. 

a  —  x:a  +  x  =3:7, 

10. 

x  :  1.5  =  lj- :  1.8. 

Properties  of  Proportions  223 

EXERCISE 
331.     Find  the  value  ofx  in  each  of  the  proportions  1  to  10. 

1.  8  :  x  =  24  :  3. 

2.  9:81  =  ^:243. 

3.  18: 7.2  =  .4:  a. 

4.  a:b  =  x:  c. 

5.  a; :  a  =  & :  c. 

6.  x  +  9  :  8  =  x :  3. 

/State  wTwcft  o/  £/*e  proportions  11  to  16  are  correct  and  which 
are  incorrect. 

11.  5:6  =  15:18.  13.    3:5  =  77:112. 

12.  2:3  =  5:8.  14.    5:7  =  10:11. 
15.    (x  +  y):(x-  y)=(x2  +  2xy  +  ?):(& - y2). 

5m  +  3  __  5m  —  3  ^ 
10m  +  9~10m-  9* 
17.   What  is  a  fourth  proportional? 

Find  the  fourth  proportional  to  each  of  the  sets  of  three  num- 
bers in  18  to  23. 

m  —  n(m  —  n)2m 

m  ■+-  n    (m  -f  n)2   n 

nn     a2-b2   1       a    1  _b 


18. 

5,  6,  10. 

19. 

8,  7,  5. 

20. 

ra,  7i,  p. 

21. 

Ill 

a   b    c 

a2  +  b2'         b  a 

24.   What  is  a  third  proportional  ? 

Find  the  third  proportional  to  each  of  the  sets  of  two  numbers 
in  25  to  29. 

25.   9,  6 ;  16,  12.  OQ         m2       Im  -  m2 


28. 


26.  (o  -  b)2,  a2  -b\  V  ~  ™2    (!  +  ™)2 

27.  P*-4*  p-qm  29.        i9  1^!_. 

r  r  (j)  -fm)2'  m?  -\-p?' 


224  ,        Ratio  and  Proportion 

2     4 

332.  Consider  the  proportion,  -7  =  5-     Clearing  of  fractions 

4     8 

gives  42  =  2  •  8  or  4  =  V2  •  8.     This   example   illustrates   the 
following  property : 

II.  A  mean  proportional  between  two  numbers  is  equal  to  the  square 
root  of  their  product. 

Proof.     Let  a,  b,  and  c  be  such  numbers  that 

a_b 

b~c' 

b2  =  ac.         (Clearing  of  fractions.) 
.-.  b  =  Vac.     (Extracting  the  square  root  of  both  members.) 

Find  the  mean  proportional  between  3  and  12. 

Solution.  3  :  x  =  x  :  12. 

x2  =  36^ 
x  =  V36,  or  6. 
This  may  be  verified  by  noting  that  3  : 6  =  6  :  12  is  a  true  proportion. 
(Why  ?) 

EXERCISE 

333.  Find    the    mean   proportional    between    each   pair    of 
numbers : 

1.  25  and  36.  4.   5  a2  and  5  62. 

2.  9  and  81.  5.    9  a  and  4  ab2. 

3.  4  a  and  ab2.  6.    3  a2b2  and  12  c2. 

8.    *- and  (™  +  5?. 

m2  +  10m  +  2o  125 

9.    Find  a  third  proportional  to  3  and  5. 

10.  Find  a  third  proportional  to  x2  —  y2  and  x  —  y. 

11.  5  a&  is  a  mean  proportional   between  15  a2  and   what 
other  number  ? 

12.  3«    is    a    mean    proportional    between   18   and   what 
number  ? 


Properties  of  Proportions  225 

334.  From  such  an  equation  as  3  •  8  =  4  •  6,  we  may  form 
proportions  by  a  proper  arrangement  of  the  numbers. 

Thus,3=6    3  =  4    8  =  6 
'4      8'   6      8'   4      3 

Can  a  proportion  be  made  from  the  numbers  involved  in 
the  equation  4  •  10  =  5  •  8  ? 

III.  If  the  product  of  two  numbers  is  equal  to  the  product  of  two  other 
numbers,  the  factors  of  either  product  may  be  made  the  means  and  the 
factors  of  the  other  product  the  extremes  of  a  proportion. 

Proof.  Let  ad  =  be. 

Dividing  both  members  of  this  equation  by  bd,  we  have 

-  —  2. 
b~ d 

Form  proportions  from  the  equation  pq  —  xy. 
Solution.  &  =  &. 

qy    qy 

.•.  2-*,orp  :y  =  x:q. 

y    q 

AlsoH  =  ^. 
px     px 

.-.  2  =  V-,  or  q  :  x  =  y  :p. 

x     p 

Let  the  student  form  proportions  by  dividing  both  members  of  pq  =  xy, 
(1)  hjpy,  (2)  by  qx. 

In  writing  a  proportion  from  two  equal  products,  if  any  one 
factor  of  either  of  the  products  is  written  as  first  term  in  a 
proportion,  the  other  factor  of  that  product  becomes  in  every  case 
the  last  term. 

EXERCISE 

335.    1.    Form  proportions  from  ad  =  be  by  dividing  both 
members  by  cd ;  by  ac ;  by  ab. 

2.   Form  a  proportion  from  2  x  =  3  y. 

Suggestion.  Divide  both  members  of  the  equation  by  2  y.  Could  a 
proportioD  be  formed  by  dividing  by  3  x  ?  by  6  ? 


226  Ratio  and  Proportion 

3.  Form  a  proportion  from  5u  =  7 w. 

4.  Form  a  proportion  from  x2  =  2  ab. 

5.  Form  a  proportion  from  #2  —  y1  =  a2  —  62. 

6.  Can  the  numbers  2,  9,  3,  and  7  be  arranged  as  the  terms 
of  a  proportion  ?  Explain.  Can  6,  8,  4  and  12  be  so  arranged  ? 
Why? 

7.  Write  a  proportion  from  a  =  be. 

8.  What  is  the  ratio  of  x  to  y  in  12  x  =  30  y  ? 

9.  What  is  the  ratio  of  x  to  y  m.2>x  —  2y  =  x~\-y? 
10.    Find  the  ratio  of  a  to  b  in 

2a-Sb  =  2c-Sd 
b  d 

336.  IV.  If  four  numbers  are  in  proportion,  they  are  in  proportion  by 
inversion  ;  that  is,  the  second  term  is  to  the  first  as  the  fourth  is  to 
the  third. 

Proof.  Let  -  =  -. 

b     d 

ad  =  bc.     (Why?) 
.-.  -  =  -.     (Dividing  by  ac.) 

23. 

Transform  -  =  -  by  inversion. 
6     9    J 

2     3 
Solution.  -  =  -. 

6     9 

6  =  9 
2     3 
Let  the  student  test  the  correctness  of  this  last  proportion. 

2     4 

337.  If  we  interchange  the  means  of  the  proportion  -=-, 
o     3  3     6 

we  get  -  =  -,  which  is  another  proportion.     This  transforma- 
tion is  always  possible,  and  is  stated  as  follows  : 

V.  If  four  numbers  are  in  proportion,  they  are  in  proportion  by  alter- 
nation ;  that  is,  the  first  term  is  to  the  third  term  as  the  second  is  to 
the  fourth. 


(Why  ?) 


Properties  of  Proportions  227 


Proof.  Let 


b     d 
ad  =  bc.     (Why?) 

,«4      (Why?) 

Transform  -  =  —  by  alternation. 
5     10    J 

Solution.  -  =  — 

5      10 

f=A.     (Why?) 

8      10       v       J    } 

ORAL  EXERCISE 

338.  Transform  the  proportions  1  to  4  by  inversion.  Trans- 
form them  by  alternation. 

1.2:3  =  6:9.  3.    3  :  -  2  =  -  9  :  6. 

2.    x  :  y  =  a  :  b.  4.    a  :  2  a  =  b  :  2  b. 

5.  Can  the  proportion  $  5  :  $  10  =  2  men  :  4  men,  be  trans- 
formed by  alternation  ?     Explain. 

6.  Can  §  330,  I,  be  applied  to  the  proportion  in  the  last 
example  ?     Explain. 

339.  Given  the  proportion  -  =  — .     From  this  we  may  make 

.,  , .  ,  n  4  +  5     8  +  10         9      18     T 

another  proportion  as  follows  :    — ! —  =  — — —   or  -  =  — - .    in 

5  10  5      10 

general  this  may  be  stated  as  follows : 

VI.  If  four  numbers  are  in  proportion,  they  are  in  proportion  by  com- 
position ;  that  is,  the  sum  of  the  first  two  terms  is  to  the  second  as  the 
sum  of  the  last  two  terms  is  to  the  fourth.  Or  the  sum  of  the  first  two 
terms  is  to  the  first  as  the  sum  of  the  last  two  terms  is  to  the  third. 

Proof.  Let   -  =  -. 

b     d 


1  +  1*1  +  1.      (Why?) 
a±b  =  c±dt     (Why?) 


228  Ratio  and  Proportion 

To  prove  2-tA^O+A  transform  the  proportion  -  =  -   bv 
a  c  b      d     J 

inversion   and   then  proceed   as  before.     Let  the  student  do 

this. 

q       f* 

Transform  by  composition  -  =  -  . 

Solution.  i+i  =  °±«  or  I  =  1* . 

4  8  4      8 

EXERCISE 

340.   1.    Given  *  =  ™,  prove  that  *  +  m  =  ^±^  ■ 
y      n  m  n 

Hint.     Apply  first  V  and  then  VI. 
2.    Transform  by  composition  x 


3.    Solve  the  equation 


5         2 

x  —  2      x  —  1 


5  —  x     3  —  x 

4.  Solve  the  equation  in  example  3,  first  transforming  by 
composition. 

5.  If  -  =  -  ,  prove  that  ^  =  *+_?.     (Use  V  and  VI.) 

b     d'  *  c  d  v  J 

6.  U  -  _  -,  prove  that  ^^+^  =  - .     (Use  VI  and  V.) 

n      y  x  +  y       y 

3      9 
341.   Given   the   proportion   -  =  — .      From   this   we   may 

q  A         Q  -jo  1  3 

make   a  proportion   .  = or  - —  =  — —  .     We    may 

F    F  4  12  4         12  J 

also  write      ~     =  — ^— ;  that  is,  -  =  — . 
4  12     '  '  4      12 

In  general  this  may  be  stated  as  follows : 

VII.  If  four  numbers  are  in  proportion,  they  are  in  proportion  by 
division ;  that  is,  the  difference  between  the  first  two  terms  is  to  the 
second  term  as  the  difference  between  the  last  two  terms  is  to  the  fourth. 
Or  the  difference  between  the  first  two  terms  is  to  the  first  as  the  differ- 
ence between  the  last  two  terms  is  to  the  third. 


Properties  of  Proportions  229 


Proof.  Let  ^=- 


b  d 

Let  the  student  complete  the  proof. 

5      10 
Transform  -  =  —  by  division. 
2       4     J 

0  5     10 

Solution.  -  =  — 

2       4 


5  - 2  _  10  -  4    or  3_6 

2  4      '        2     4' 


EXERCISE 


342.    1.   If  -  =  -,  prove  that  ^j^c-d        (Apply   IV 
b     d  a  c 

and  proceed  as  above.) 

2.  Apply  the  transformation  by  division  to  a  ~*~     =  c  "*"     • 

3.  Apply    the    transformation    by    composition    to    — - — 
c  —  d 


d 

4.  If  ^L±l  =  &   find  the  value  of  ™. 

no  w 

5.  Solve  £±i  =  ?.     (Apply  VII.) 

343.  The  last  two  transformations  are  sometimes  referred 
to  as  transforming  a  proportion  by  addition  instead  of  by 
composition,  and  by  subtraction  instead  of  by  division. 

344.  A  combination  of  the  two  preceding  transformations 
may  be  made. 

Thus,   §  =  ^,and3-±^  =  ^±^or-^=*L. 
5      15  3-5     9-15        -2      -6 

This  illustrates  the  following  property  of  a  proportion : 


230  Ratio  and  Proportion 

VIII.  If  four  numbers  are  in  proportion,  they  are  in  proportion  by 
composition  and  division ;  that  is,  the  sum  of  the  first  two  terms  is  to  their 
difference  as  the  sum  of  the  last  two  terms  is  to  their  difference. 


Proof. 


Let": 
b 

_  c 
~d' 

a  +  b 

c  +  d 

b 

d 

a  —  b 

_c-d 

(Why?) 


Also  - -  =  Z .     (Why  ?^> 

b  d  V       J    / 

Dividing  the  last  two  equations  member  by  member,  we  have 

a  -\-  b  __c  +  d ^ 

a  —  b      c  —  d 

p*      10 

Transform     =  — -  by  composition  and  division. 
6     12 


Solution. 

5_10 
6      12* 

5  +  6  _  10  +  12 

5-6      10-12' 

11  _  22 
-1      -2* 

EXERCISE 

345.  1.  Transform 4  :2  =  12  :  6  by  composition  and  division. 

2.  Transform  — ——  =  — i__  by  composition  and  division. 

a-  2      ra-3     J         F 

3.  a:b  =  c  +  x:  c  —  x.     Solve  for  x,  using  §  330,  I. 

4.  Solve  the  equation  in  3,  using  §  344,  VIII. 

5.  If  2  =  -c,  show  that  ?—b-  =  £^. 

b      d  a  +  b     c  +  d 

6.  If    -  =  -,  show   that   ±±c  =  'b±A.     (Alternation  and 

b     d  a—  c     b  —  d 

composition  and  division.) 

346.  If  several  fractions  are  equal  to  each  other,  the  sum  of 
their  numerators  divided  by  the  sum  of  their  denominators 
equals  any  one  of  the  fractions. 


Properties  of  Proportions  231 

Thus,  ?  =  §  =  1  =  ?!  and  2  +  6  +  8  +  14  or  §5  is  equal  to  any  one 
'  3     9      12     21  3+9  +  12  +  21        45        H  * 

of  these  fractions. 

This  property  of  equal  fractions  may  be  stated  thus  : 

IX.   In  a  series  of  equal  ratios  the  sum  of  the  antecedents  is  to  the 
sum  of  the  consequents  as  any  antecedent  is  to  its  consequent. 

.~  T    ,  a      c     e     x 

Proof.  Let  T  =  -  =  -  =  -  • 

b     d     f     y 

Also  let  each  ratio  equal  k. 

-  =  k,  from  which  a  =  bk.     (Why  ?) 

-  =  k,  from  which  c  =  dk. 
d        ' 

-=k,  from  which  e  =  fk. 

-  =  k,  from  which  x  =  yk. 

y 

Then  a  +  c  +  e  +  x  =  k(b  +  d  +  /+  y). 

a-|-  c  +-  e  +  #      _       a      c     e     x 
k 


b+d+f+y  b     d     f     y 

rhus>l  =  2  =  3^4  =  l+2  +  3  +  4Qr10i 

'2      4      6     8      2+4  +  6  +  8       20 


EXERCISE 

347.    1.   Apply  IX  to  ^=1  =  ^. 

2.  Apply  IX  to  £  =  £. 

3.  If  -  =  -  =  -  =  -,  what  is  the  value  of  a  +  c  +  e? 

b      d     f     3'  b  +  d+f 

.     To  a     m      x     i        , ,    ,  a  —  m  +-  a;      a 

4.  It  -  =  —  =  - ,  show  that  ■ —  =  -  • 

b      n      y  b  —  n  +-  y      b 

Hint.     —  may  be  replaced  by  ^^  •     (Why?)     Then  apply  IX. 
n  —n 


232  Ratio  and  Proportion 

5.  If  °  =  °=g,  show  that  j»q  +  3c  +  4r  =  ». 

b     d     s'  26  +  3d  +  4s     b 

Hint.      fl-  =  2-a. 
b     26 

6.  If    = %- = ,  prove  that  each  one 

a-\-b  —  c     a  —  b  +  c     b  +  c  —  a 

of  these  fractions  is  equal  to       '  ^  "*   -» 

a  +  6  +  c 

348.    1  =  ?    also  i-2  =  -  or  -  =  ^-     From  ?  =  *  we  may 
3      6'  32     62         9     36  3     6  J 

get  -  =  —  by  squaring  both  members  of  the  equation. 
9      oo 

These  examples  illustrate  the  following  property  : 

X.   If  four  numbers  are  in  proportion,  the  squares  (or  any  like  powers) 
of  these  numbers  are  in  proportion. 

Proof.  Let  -  =  -  • 

b      d 

Squaring  both  members  of  this  equation,  we  have 

l2 


(t)'-(S 


"  b2      d2' 
The  proof  for  other  like  powers  is  similar. 

Thus,  -  =  i-  •     How  does  it  follow  that  —  =  —  ? 
'6      10  25     100 

EXERCISE 

349.     l.    If  —  =  -,  prove  that  —  =  —  • 
n      y  x2      y2 

2.  If  %  =  - ,  show  that  ~-^  =  %'     (x  and  IX-) 

b      s'  b2  +  s2      b2       v  ' 

a2       c2 

3.  In  the  proof  of  X  we  produced   the  equation  —  =  — 

a      c 


from  -  =  -•     Is  —  =  -?     Explain. 
b     a  ol      o 


Summary  of  the  Properties  of  Proportions     233 


SUMMARY   OF   THE   PROPERTIES   OF   PROPORTIONS 

350.   Following  are  statements,  in  algebraic  symbols,  of  the 
properties  of  proportions : 


II.   If  a:  b=  b:c,  then  b  =  Vac. 

III.  If  ad  =  be,  then  a.b=c:d  etc 

IV.  If  a:  b  =  c:  d,  then  b:a  =  d:  c. 
V.    If  a:  b  =  c:  d,  then  a:c=  b:d. 

VI.    If  a:  b  =  c:d,  then  a  +  b:  b  =  c+  did. 
Or  a  +  b:a  =  c  +  d:  c. 
VII.    If  a:  b=c:d,  then  a  -  fc:  &  =  c  -  d:  rf. 
or  a  —  b:a=  c  —  d:  c. 
VIII.    If  a:  b  =  c: d,  then  a  +  b:  a  -  b  =  c  +  d:  c  -  d. 
IX.   If  a:b=  c:d=  e:f,  then  a  +  c+  e:  b  +  d  +  f=  a:b. 
X.    If  a :  &  =  c :  d,  then  d2 :  b2  =  c2 :  d2,  or  a"  :  b"  =  V :  d". 

EXERCISE 

351.  1.  What  is  meant  by  transforming  a  proportion  by 
inversion  ?  by  alternation  ?  by  composition  ?  by  division  ?  by 
composition  and  division  ? 

2.  If -  =  -,  show  that  —=^;   also  that— =  —• 

b      d'  5b      5d  2b     2d 

3.  If  «  =  £,  show  that  2q  +  56  =  2c  +  5cl. 

6      d'  2a-5b     2c-5d 

4.  Apply  I  to  see  if  13  :  17  =  19  :  24. 

5.  Given  «  =  £,  show  that  «+26  =  c  +  2d. 

6      rf  a  c 

6.  Find  a  mean  proportional  between  SB-L.  and  2-^JL. 

p-q         P  +  <1 

7.  If  £=4=4  show  that  ^t_C=^;  also  that  *±±  =  *. 


234  Ratio  and  Proportion 

8.  If^  =  ^  =  e,showthataJ--c  =  ^±-e  =  c+^. 

b      d    f  b+d     b+f     d+f 

9.  (a)  Find  a  third  proportional  to  —  and  -• 

lb  b 

(b)  Find  a  fourth  proportional  to  a3  —  b3,  a2  —  b2,  a  —  6. 

10.  If  £  =  £  and  g- ^,  show  that  ™'  =  ^. 

6      d  b'     d'  bb'      dd' 

11.  Transform  so  that  x  shall  occur  only  once,  -  =  c-i-®. 

b         x 

Solution.  <*  =  <L±J5. 

b         x 

a  —  b  _  c  +  x  —  x  Qr  a  —  b  _  c 


x 


12.    Transform  so  that  x  shall  occur  only  once : 
(«)*±2=|.  (c)      • 


x         3  6  + «     6  —  a; 

(6)  ?=«±».  (d)   «=     K 


b      c  —  x  b      x  —  c 

13.  A  cement  block  is  to  be  made  of  Portland  cement,  sand, 
and  gravel  in  the  proportions  1:2:3.  How  much  of  each  is 
there  in  a  block  that  weighs  300  pounds  ? 

14.  The  unequal  sides  of  a  rectangle  are  in  the  ratio  of  3  to 
5.     Find  the  dimensions  if  the  perimeter  is  20  feet. 

15.  If  r^^  =  -,  find  the  value  of  ^. 

m  —  n      3  n 

16.  What  number  added  to  each  of  the  numbers  1,  3,  19, 
and  27  will  give  numbers  that  form  a  proportion  ? 

17.  A  and  B  do  a  piece  of  work  for  %  38.  A  works  5  days 
of  8  hours  each  and  B  works  4  days  of  9  hours  each.  How 
should  the  money  be  divided  ? 

18.  The  angles  of  a  certain  triangle  are  in  the  ratio  1:2:3. 
Their  sum  equals  180°.     How  large  is  each  ? 


Summary  of  the  Properties  of  Proportions     235 


19.  The  sides  of  a  triangle  are  in  the  ratio  3:4:5.     The 
perimeter  is  100  inches.     How  long  is  each  side  ? 

20.  li  m  -\-  n:m  —  n  =  x  +  y  :x  —  y,  show  that 

x2  +  m2 :  x2  —  m2  =  y2  +  n2 :  y2  —  n2. 

For  what  value  of  x  does  each  set  of  numbers  form  a  true  pro- 
portion if  taken  in  the  order  given  ? 

21.  3,  4,  5,  x.  25.   5,  6,  3  +  x,  4  +  x. 

22.  2,  3,  x  +  1,  x  +  2.  26.   15  +  x,  20  +  x,  1,  6. 

23.  a;  +  1,  a?  +  2,  a;  +  4,  a;  +  8.      27.    3  4-  a,-,  4  4-  x,  25,  32. 

24.  m  +  n,  x—l,  m  —  n,  x  +  1.      28.    a;,  121  —  x,  5,  6. 

29.  Find  the  ratio  of  x  to  y  if  4a?~3y  =  3. 

u       2x  +  5y 

30.  Show   that   four   consecutive   numbers   cannot  form  a 
proportion. 

Hint.     Let  n,  n  +  1,  »  +  2,  n  +  3  represent  the  numbers. 

31.  Brass  consists  of  2  parts  of  copper  to  1  part  of  zinc. 
How  many  pounds  of  each  are  there  in  9  pounds  of  brass  ? 

32.  Gunmetal  consists  of  9  parts  of  copper  to  1   part   of 
tin.     How   many   pounds   of   each  are 
there  in  20  pounds  of  gunmetal  ? 

33.  It  is  proved  in  geometry  that  if 
a  line  is  parallel  to  one  side  of  a  tri- 
angle, it  divides  the  other  two  sides  into 
parts  that  are  in  proportion.  By  actual 
measurement  in  the  figure  show  that  I :  m  =  n :  r. 

34.  If  I  =  5  inches,  m  =  2  inches,  and  n  =  4  inches,  find  r. 

35.  If  I  =  8  inches,  m  =  3  inches, 
and  r  =  2.5  inches,  find  n. 

36.  To  measure  the  width  of  a  river, 
BC,  a  triangle  was  laid  out  as  shown 
in  the  figure,  with  BD  parallel  to  EC. 
By  actual  measurement  AB  was  found 

to  be  96  feet,  AD  was  76  feet,  and  DE  was  102  feet.     Find  BC. 


236 


Ratio  and  Proportion 


37.  Similar  triangles  are  triangles  that  have  the  same  shape. 
It  is  stated  in  geometry  that  their  corresponding  sides  are  in  pro- 
portion.    Thus  in  the  two  triangles  l:p=m:q;  also  n:r=m:q. 

Write  another  proportion  involving  the  sides  of  the  triangles. 


38.  If  I  =  5  inches,  m  =  3.8  inches,  srndp  =  7  inches,  find  the 
length  of  q.     Also  find  the  length  of  n  if  r  —  8  inches. 

39.  Of  two  similar  triangles  (see  example  37)  the  sides  of 
one  are  5  inches,  8  inches,  10  inches,  and  the  sides  of  the  other 
are  1\  inches,  12  inches,  15  inches.  Show  that  their  perime- 
ters are  in  the  same  ratio  as  two  corresponding  sides. 

40.  In  the  figure,  XY  is  the  length  of  the  shadow  of  the  tree, 
TZ  is  the  height  of  the  tree  ;  BC  is  a  stick  set  in  the  ground 

and  AB  the  length  of  its 
shadow.  If  AB  =  6  feet, 
BC=4±  feet,  and  XF=42 
feet,  find  the  height  of  the 
tree. 

Hint.    The  triangles  are  sim- 
ilar.    (See  example  37.) 

41.  The  Woolworth 
building  (city  of  New 
York),  the  highest  office 
building  in  the  world,  casts  a  shadow  625  feet  long  at  the 
same  time  that  a  boy  4.8  feet  high  casts  a  shadow  4  feet  long. 
How  high  is  the  building  ? 


XIIL    GRAPHS1 

352.  The  student  has  seen  in  his  general  reading  many  dif- 
ferent graphical,  or  pictorial,  methods  of  representing  data. 
A  series  of  straight  lines  can  be  used  to  show  the  relative 
values  of  the  grain  crops,  manufactured  products,  or  the  wealth 
of  different  countries.  Pictures  of  soldiers  of  different  sizes 
may  represent,  pictorially,  the  relative  strength  of  the  armies 
of  different  nations.  In  a  similar  way  the  strength  of  navies 
may  be  represented  by  ships. 

353.  1.  Determine  by  construction  a  line  representing  the 
sum  of  three  given  lines  a,  b,  c. 

Place  the  three  lines,  a,  b,  and  c  end  to  end  and  the  total 
length  is  the  sum  required. 


2.  Represent  on  a  scale  of  \  inch  to  a  mile  a  distance  of  10 
miles. 

The  line  AB  is  10  units  long,  each  unit  being  \  inch.  AB 
therefore  represents  a  distance  of  10  miles  on  a  scale  of  \ 
inch  to  a  mile. 

A  i i i i t        i i i i i        iR 

3.  Represent  a  distance  of  25  miles  on  a  scale  of  i  inch  to  a 
mile. 

1  The  chapter  on  graphs  may  he  omitted,  if  desired,  without  interrupting 
the  sequence  of  the  work. 

237 


238 


Graphs 


4.  The  beet  sugar  produced  in  the  United  States  from  1901 
to  1911  expressed  in  tons  was  as  follows:  1901,  184,000; 
1903,  240,000;  1905,  313,000;  1907,  464,000;  1909,  512,000; 
1911,  606,000. 

By  using  a  distance  of  \  inch  to  represent  100,000  tons  these 
facts  may  be  represented  graphically  as  follows : 


1901 

1903 

1905 

1907 

1909 

1911 

( 

) 

I 

- 

1 

i 

6 

100,000  Tons 

5.  In  the  following  table  used  by  life  insurance  companies 
the  premium  charged  a  person  at  the  age  of  30  is  computed 
on  the  basis  that  he  is  expected  to  live  35.33  years.  Illustrate 
graphically  the  expectation  of  life  for  ages  from  10  yr.-  to  80  yr. 

Draw  two  straight  lines  perpendicular  to  each 
other.  Measure  off  on  each  equal  spaces  repre- 
senting age  and  expectation  of  life,  allowing  4 
spaces  for  each  10  years.  The  expectation  of  life 
for  10  years,  namely,  48.72  years,  is  shown  at  E\. 
In  the  same  way  E2,  E3,  etc.  may  be  located, 
showing  the  expectation  of  life  at  20  yr.,  30  yr., 
etc.  of  age.  A  continuous  curve  drawn  through 
these  points  is  the  expectation  of  life  curve  for 
ages  from  10  yr.  to  80  yr.  This  curve  shows  at  a 
glance  the  expectation  of  life  for  all  ages  from  10 
yr.  to  80  yr. 

It  is  possible  to  estimate  from  this  curve  the  expectation  of  life  for 
ages  not  given  in  the  table. 


Age 

Expectation 
of  Life 

10 

48.72 

20 

45.5 

30 

35.33 

40 

28.18 

50 

20.91 

60 

14.1 

70 

8.48 

80 

4.39 

Graphs 


239 


Determine  from  the  figure  the   expectation  of   life  at   the 
ages  15,  25,  35,  55,  75. 


« 

£, 

0 

;  •' 

30 

a 

'■& 
as 

» 

| 

ffj 

» 

*v 

10 

k[ 

> 

10 


20 


.30 


40  50 

Age 


oo 


70 


6.    The   following   table   shows   the   annual    premium    per 
$  1000  at  different  ages  for  life  insurance. 


Age 

21 

25 

30 

35 

40 

45 

50 

55 

60 

Premium 

$  18.40 

$  20.14 

$  22.85 

$  26.35 

$30.94 

$37.08 

$45.45 

$  56.93 

$72.83 

Construct  a  curve  showing  the  relation  between  the  age  and 
the  premium.  Measure  the  ages  along  the  horizontal  line  and 
the  premium  on  the  vertical  line. 

Note.     The  pupil  should  use  cross-section  paper  for  this  work. 
From  the  curve  estimate  the  premium  for  a  person  at  the  age  of  28,  37, 
42,64. 

7.  The  following  temperatures  were  taken  from  the  weather 
reports  at  a  certain  city  for  January  and  February. 


240 


Graphs 


Day 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

January 
February 

30° 

24° 

31° 

28° 

32° 
36° 

32° 

24° 

26° 

22° 

26° 
32° 

31° 
23° 

34° 

4° 

34° 

6° 

25° 
11° 

20° 

7° 

Day 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

January 
February 

16° 
0° 

9° 
6° 

20° 
16° 

34° 
13° 

38° 

14° 

28° 
18° 

26° 

22° 

32° 
20° 

36° 
16° 

24° 
18° 

20° 
17° 

Day 

23 

24 

25 

26 

27 

28 

29 

30 

31 

January 
February 

34° 

4° 

34° 

7° 

18° 
14° 

33° 

25° 

42° 
34° 

42° 
36° 

44° 

30° 

28° 

This  temperature  record  is  shown  graphically  in  the  follow- 
ing figure : 


Iff 

j 

inuar 

y 

»S 

, 

F 

■bi 

ua 

ry 

} 

I 

^C 

7 

/ 

\ 

<, 

ft 

> 

i 

\ 

/ 

\ 

/ 

I 

k 

¥ 

\ 

I 

1"/ 

\ 

i 

^ 

\ 

( 

i 

/ 

\ 

V 

s 

J 

1 

H 

n° 

\ 

T 

\ 

/ 

', 

: 

3 

1 

■ 

1 

)  1 

0   1 

1  1 

2  1 

?,  i 

i  l 

j  l 

;  i 

7   1 

8   1 

'J  i 

0  I 

1  2 

2  2 

3  2 

L  2 

r)  2 

1  2 

7  2 

8  2 

i  S 

o  a  i 

Observe  that  time,  or  dates,  are  represented  on  the  horizontal 
line  using  1  space  for  one  day ;  the  temperatures  are  measured 
in  the  direction  of  the  vertical  line,  using  1  space  for  5°. 


Graphs 


241 


8.  Two  trains  leave  Chicago  going  east  on  parallel  lines. 
One  starts  at  noon  and  runs  at  the  average  rate  of  30  miles  an 
hour ;  the  other  starts  at  1  o'clock  and  runs  40  miles  an  hour. 
How  far  from  Chicago,  and  at  what  time,  will  the  fast  train 
overtake  the  slow  train  ? 

Let  the  spaces  on  OY  represent  the  number  of  miles  traveled 
as  indicated,  and  the  spaces  on  OX  represent  the  time. 


y 

y 

/ 

/ 

S 

jt 

$ 

y 

a 

t 

p< 

0D 

p* 

80 

p-, 

n 

\ 

40 
D 

... 

... 

... 

fl 

Ft 

h 

n\s 

X 

o 

12 

i 

i 

i 

i 

Hours 


At  1  o'clock  the  slow  train  will  have  run  30  miles.  Measure  the  30 
miles  along  OY  as  OB.  Measure  the  time,  1  hour,  along  OX.  Draw 
the  rectangle  ODP\Pb.  Pi,  by  its  distance  from  OX,  represents  the 
distance  traveled,  and,  by  its  distance  from  OF,  represents  the  time. 
Similarly  P2  represents  the  distance  and  the  time  after  2  hours; 
P3,  after  3  hours,  etc.  The  points  O,  Pi,  P2,  P3..-lie  in  a  straight 
line.  Draw  this  line  and  call  it  h.  If  any  point  is  taken  on  this  line,  it 
will  be  found  that  a  distance  and  the  corresponding  time  can  be  read  at 
once  from  the  figure.  In  a  similar  way  draw  Z2  through  P6P6P7P4, 
representing  the  progress  of  the  fast  train. 

It  is  evident  that  the  intersection  of  the  lines  h  and  l2  will  indicate 


242 


Graphs 


the  time  of  the  day  and  the  distance  traveled  when  the  distances  are 
equal ;  that  is,  when  the  fast  train  overtakes  the  slow  train.  From  the 
figure  it  appears  that  this  occurs  at  4  o'clock  when  the  trains  are  120 
miles  east  of  Chicago. 

Determine  from  the  figure  how  far  the  fast  train  is  behind 
the  slow  train  at  3  o'clock.  When  will  the  fast  train  pass  the 
point  where  the  slow  train  was  at  2  o'clock  ? 

354.  In  representing  statistics  and  data  graphically,  first 
look  over  the  numbers  involved  so  as  to  choose  convenient 
units.  In  general,  if  the  numbers  are  large,  select  small 
units. 

EXERCISE 

355.  1.  If  a  person  saves  10^  a  day  and  deposits  it  in  a 
savings  bank  which  pays  3  %  interest,  the  balances,  to  the 
nearest  dollar,  at  the  end  of  certain  years  are  as  follows : 


Year 

1 

2 

3 

5 

8 

10 

14 

17 

20 

Balance 

$37 

$75 

$115 

$197 

$  330 

$425 

$635 

$809 

$999 

Using  two  spaces  on  the  horizontal  line  OX  for  one  year, 
and  four  spaces  on  O  Y  to  represent  $  100,  draw  a  smooth 
curve  through  the  points  located  from  the  table  and  estimate 
the  balances  for  the  years  omitted. 

2.  The  table  below  gives  the  expense  and  receipts  of  a 
certain  newspaper  for  various  numbers  of  copies. 


Numbers  of  copies 

1000 

2000 

3000 

4000 

Expense  in  dollars 

425 

550 

675 

800 

Receipts  in  dollars 

300 

495 

690 

885 

Graphs 


243 


Construct  a  graph  showing  the  relation  between  the  number 
of  copies  produced  and  the  expenses.  On  the  same  diagram 
show  the  relation  between  the  receipts  and  the  number  of 
copies.  From  the  diagram  estimate  as  nearly  as  possible  the 
smallest  number  of  copies  that  can  be  produced  to  make  the 
paper  pay.  Use  1  inch  on  the  horizontal  line  for  1000  copies, 
and  \  inch  on  the  vertical  line  to  represent  $  100. 

3.  The  following  table  shows  the  distances  in  miles  of 
certain  railway  stations  from  Chicago,  and  the  time  of  two 
trains,  one  to  and  one  from  Chicago.  If  each  run  is  to  be 
made  at  a  constant  speed,  show  graphically  the  progress  of 
each  train. 


Going  West 

Miles 

Chicago 

Miles 

Going  East 

9  :00  a.m. 

0 

284 

6:00 

12  :  40  p.m. 

127 

arrive 

Bloomington 

leave 

157 

1  :30 

12:45 

leave 

arrive 

1  :25  p.m. 

2:25 

185 

arrive 

Springfield 

leave 

99 

12  :  00  m. 

2:35 

leave 

arrive 

11  :  55 

4:45 

258 

Alton 

26 

10:00 

5:45 

284 

St.  Louis 

0 

9  :  00  a.m. 

At  what  point  do  the  trains  pass  and  how  far  is  each  from 
Chicago  ? 

Let  the  horizontal  line  represent  the  time,  using  one  half  inch 
for  one  hour  and  one  inch  on  the  vertical  line  for  100  miles. 

4.  The  following  table  gives  the  length  of  the  circumferences 
of  a  circle  for  given  radii : 


Radius 

0 

1 

2 

4 

6 

8 

10 

Circumference 

0 

6.28 

12.56 

25.12 

37.7 

50.24 

62.8 

Measure   the   circumferences   along   the   vertical   axis,   us- 
ing  one   space   for  2  units.     Use  two  spaces  for  1  unit  in 


244 


Graphs 


measuring  radii  on  the  horizontal  axis.  Locate  all  the  points 
tabulated  and  draw  a  smooth  curve  through  them,  (a)  Estimate 
from  the  figure  the  circumference  of  a  circle  with  radius 
2-j-  units,  (b)  What  is  the  approximate  radius  of  a  circle  whose 
circumference  is  44  units  ? 

5.   The  following  table  shows  the  areas  of  circles  for  certain 
radii : 


Radius 

0 

1 

2 

3 

4 

5 

AY-ea 

0 

3.14 

12.56 

28.26 

50.24 

78.5 

Locate  the  points,  using  the  same  units  as  in  the  last  example. 
Draw  a  smooth  curve  through  the  points. 

(a)  Estimate  the  area  of  a  circle  with  radius  2\  inches ; 
6  inches.  (6)  Estimate  the  radius  of  a  circle  with  an  area  of 
40  square  inches  ;  of  70  square  inches. 

356.  Axes  and  Coordinates.  If  we  draw  two  straight  lines 
at  right  angles  to  each  other  as  in  the  figure,  we  divide  a 

plane  surface  into  four 
quadrants.  The  lines 
are  the  axes.  The  hori- 
zontal line  XX'  is  the 
X-axis,  and  the  verti- 
cal line  77'  is  the 
Y-axis.  The  quadrants 
are  the  first  quadrant, 
the  second  quadrant, 
the  third  quadrant,  and 
the  fourth  quadrant,  as 
indicatedbytheBoman 
notation.  We  name  the 
spaces  along  these  axes 
as  shown  in  the  figure. 
If  we  select  any  point  in  the  plane,  as  Plt  we  can  describe 
its  position  completely  by  telling  how  far  it  is  to  the  right  of 


Y 

I 

\X 

+ 

3 

J 

IX 

V 

+ 
-j- 

1 

J,( 

-I 

3) 

9 

n 

Pi 

[(3, 

2) 

I 

.Y' 

Pi 

X 

( 

t 

I 

1 

0 

1 

2 

i 

I 

i 

\ 

-1 

Ft 

H 

-O 

] 

ii 

X- 

y- 

IV  f, 

+ 

-8 

1 

*< 

-2 

i-3 

) 

Ps 

-4 

Y' 

Graphs  245 

the  Y-axis  and  how  far  it  is  above  the  X-axis.  These  distances 
are,  for  the  point  Px,  3  and  2  respectively,  and  they  are  the 
coordinates  of  Px.  The  coordinate  measured  in  the  direction  of 
the  X-axis  is  the  abscissa,  usually  designated  by  x,  and  the 
coordinate  measured  in  the  direction  of  the  Y-axis  is  the 
ordinate,  designated  by  y.  The  coordinates  of  a  point  are 
written  in  the  form  (x,  y),  the  abscissa  always  being  written 
first,  followed  by  the  ordinate.  The  signs  -f  and  —  indicate 
the  direction  to  be  measured. 

EXERCISE 

357.  1.  The  coordinates  of  the  point  P2  are  (—  1,  3)  ;  of  P3, 
(-2,  -3).  What  are  the  coordinates  of  P4?  of  P6?  of  the 
intersection  of  the  axes  ? 

2.  What  are  the  coordinates  of  P6  ?  of  P7  ?  of  P8? 

3.  Locate  the  points  (-  1,  1),  (-  3,  0),  (0,  -  3),  (0,  0). 

4.  Where  are  all  the  points  which  have  abscissa  1  ? 

5.  Where  are  all  the  points  which  have  ordinate  —  2  ? 

6.  Give  the  signs  of  the  coordinates  for  each  quadrant. 

7.  How  many  points  may  have  3  and  4  as  the  absolute 
values  of  the  coordinates? 

8.  Locate  the  points  (3,  4),  (3,  2),  (3,  0),  (3,  -  1),  (3,  -  4). 
Draw  a  line  through  these  points.  What  kind  of  line  does 
this  give? 

9.  If  the  abscissa  is  zero,  where  must  the  point  be  located  ? 
10.    If  the  ordinate  is  zero,  where  must  the  point  be  located  ? 

358.  Function.  When  one  quantity  depends  upon  another 
for  its  value,  the  first  quantity  is  a  function  of   the  second. 

If  a  train  travels  at  a  uniform  rate,  the  distance  traveled 
is  a  function  of  the  time.     The  cost  of  10  yards  of  cloth  is  a 


246 


Graphs 


function  of  the  price  per  yard.  The  area  of  a  square  is  a 
function  of  its  side ;  the  area  of  a  circle  is  a  function  of  its 
radius.  The  algebraic  expression  2  x  -f-  3  is  a  function  of  x. 
In  the  equation  y  =  2x  +  3,  y  is  a  function  of  x. 


359.  Graph  of  a  Function, 
equation  y  =  2  x  +  3  may 
be  pictured  by  means  of  a 
graph.  Tabulating  sets  of 
values  of  x  and  y  that  sat- 
isfy the  equation  (allow- 
ing two  squares  for  each 
unit)  we  have  the  follow- 
ing: 


The  values  of  x  and  y  in  the 


X 

y 

0 

3 

1 

5 

2 

7 

-1 

1 

-2 

-1 

-3 

-3 

Y 

7 

M 

o 

B 

- 

R 

1 

A'' 

A' 

-4 

-. 

1 

1  / 

l 

(J 

l 

z 

! 

5 

-1 

-a 

/ 

/ 

-8 

/ 

N 

/ 

Y1 

When  all  the  pairs  of 
values  of  x  and  y  are  used  as  coordinates  we  have  a  series  of 
points  that  appear  to  lie  in  a  straight  line.  If  fractional  values 
of  x  are  taken,  other  points  between  these  will  be  found.  The 
line  MNj  if  extended  indefinitely  in  both  directions,  is  the 
graph  of  the  function  of  x,  2  x  +  3,  or  of  the  equation  y  —  2  x  +  3- 
By  this  we  mean  that : 

1.  Every  pair  of  values  of  x  and  y  that  satisfies  the  equation  will,  if 
used  as  coordinates  of  a  point,  give  a  point  on  this  line  MN. 

2.  The  coordinates  of  any  point  on  this  line  satisfy  the  equation. 


Graphs 


247 


EXERCISE 
360.   Answer  questions  1  to  A  by  referring  to  the  figure  of 
§  359,  and  verify  the  answers  by  seeing  if  they  satisfy  the  equation 
y  =  2  x  +  3. 

1.  What  is  the  value  of  x  f or  y  =  0  ?     for  y  =  —  4  ? 

2.  What  is  the  value  of  y  for  x  =  2\  ?     for  x  =  —  %? 

3.  Does  the  point  (—  3.5,  —  4)  lie  on  the  line  MN? 

4.  Do  the  values  x  =  \,  y  =  4  satisfy  the  equation  ? 

5.  W^hen  sugar  is  6  ^  a  pound,  the  cost,  c,  is  a  function  of 
the  number  of  pounds,  p.  The  equation  connecting  them  is 
c  =  6p. 

Construct  the  graph  for  finding  the  cost,  using  the  axis  OX 
for  the  cost  and  O  Y  for  the  weight. 


y 

9 

8 

H3  6 

&* 

3 
2 

1 

0 

t 

1 

0 

1 

5 

2 

0 

2 

5 

3 

0 

3 

5 

1 

0 

4 

B 

5 

0 

5 

5 

G 

0 

J 

P 

c 

0 

0 

1 

6 

3 

18 

6 

36 

10 

60 

Cents 


Note.  When  no  negative  numbers 
are  to  be  used,  the  points  are  all  in 
the  first  quadrant  and  the  bottom  line 
and  the  line  at  the  left  side  may  be  used 


Determine  from  the  figure  the  cost  of  5  pounds  of  sugar ; 
9  pounds.  How  many  pounds  can  be  bought  for  48y?  for 
55^? 


248  Graphs 

6.  Construct  a  graph  to  show  the  cost  of  eggs  at  28^  a 
dozen.  Extend  the  graph  to  8  dozen  and  estimate  from  it  the 
cost  of  2\  dozen  ;  of  5  dozen. 

Use  4  spaces  for  1  dozen  eggs  on  the  X-axis,  and  1  space 
for  7  ^  on  the  F-axis. 

7.  A  train  travels  uniformly  45  miles  an  hour.  Construct 
a  graph  and  determine  the  distance  it  covers  in  12  minutes, 
and  the  time  it  takes  to  go  24  miles. 

Hint.  The  equation  is  d  =  45  t  where  d  represents  the  distance  in 
miles  and  t  the  time  in  hours.  Use  1  space  for  six  minutes  on  the 
X-axis  and  1  space  for  5  miles  on  the  F-axis. 

8.  To  change  Fahrenheit  temperatures  to  centigrade,  the 
equation  C=  -J(F  —  32)  is  used.  In  this  equation  F  represents 
the  number  of  degrees  Fahrenheit  and  G  the  same  tempera- 
ture measured  by  a  centigrade  thermometer. 

Thus,  50°  Fahrenheit  is  changed  into  centigrade  by  substituting  50 
for  F-  C=  f(50-82)  =  |  •  18  =  10. 

Plot  the  graph  for  C  =  ^(F  —  32)  for  the  following  Fahrenheit  tem- 
peratures:   -  10°,  -  20°,  32°,  40°,  50°,  60°,  70°,  80°,  90°. 

What,  temperature  Fahrenheit  will  correspond  to  20°  centi- 
grade ?  to  25°  ?  15°  F.  corresponds  to  what  temperature 
centigrade?    72°  F.  ? 

9.  Knowing  that  1  kilogram  =  2.2  pounds,  construct  a 
graph  that  will  enable  you  to  convert  pounds  into  kilograms 
or  kilograms  into  pounds.  The  equation  is  K=2.2p.  From 
this  graph  determine  the  number  of  kilograms  in  11  pounds ; 
in  14.3  pounds;  in  22  pounds.  Determine  the  number  of 
pounds  in  3  kilograms ;  in  5  kilograms ;  in  8  kilograms. 

10.  Given  that  1  inch  =  2.54  centimeters,  construct  a  graph 
by  means  of  which  inches  can  be  converted  into  centimeters 
and  centimeters  into  inches.     The  equation  is  i  =  2.54  c. 

11.  At  noon  a  boy  begins  to  walk  along  a  road  at  4  miles 
an  hour,  and  at  2  p.m.  a  cyclist  rides  after  him  at  10  miles  an 
hour.     Show  in  a  graph  the  distance  traveled  in  any  time  by 


Graphs  249 

the  boy  and  by  the  cyclist  and  use  the  graph  to  find  when  the 
cyclist  overtakes  the  boy.     (See  example  8,  §  353.) 

12.  A  newsboy  sells  papers  at  1  cent  each  and  makes  ^  cent 
profit  on  each  paper.  Represent  graphically  his  sales  and 
profits  up  to  50  sales. 

Hint.  For  locating  points,  use  numbers  of  sales  that  are  multiples  of 
3,  as  6,  12,  18,  etc.     The  equation  is  p  =  |  s. 

13.  Another  newsboy  sells  papers  at  1  cent  each  and  gets 
-J-  cent  profit  on  each  paper.  He  has  6  cents  carfare  to  pay. 
Represent  on  the  same  axes  and  to  the  same  scale  as  used  for 
the  last  exercise  the  sales  and  profits  up  to  50  sales. 

Hint.     The  equation  is  p  =  I j  —  6. 

For  what  number  of  sales  will  the  profits  of  the  two  boys 
be  the  same  ? 

When  will  the  first  boy  make  more  than  the  second  ?  When 
will  the  second  make  more  than  the  first  ? 


XIV.    LINEAR  SIMULTANEOUS  EQUATIONS 
WITH  TWO  UNKNOWN  NUMBERS 


361.   Consider  the  equation 

x  +  y  =  5,  (l) 

where  both  x  and  y  are  unknown  numbers.    There  is  an  indefinite 
number  of  pairs  of  values  of  x  and  y  that  satisfy  the  equation. 

Thus  x  =  1,  y  =  4  is  a  solution,  since  1  +  4  =  6.     Also  x  =  2,  y  =  3  is 
a  solution,  since  2+3  =  5,  and  x=  —4,  y  =  9  is  a  solution,  since  —  4  +  9  =  5. 

Tabulating  some  of  the  values  of  x  and  y  that  satisfy  the 
equation,  we  have  the  following : 


This  tabulation  could  be  continued  indefinitely 
in  both  positive  and  negative  numbers,  and  also  in 
fractions.  This  means  that  there  is  an  indefinitely 
large  number  of  pairs  of  values  of  x  and  y  which 
satisfy  the  equation.  The  equation  is  therefore 
indeterminate. 


X 

y 

x  +  y 

1 

4 

5 

2 

3 

5 

3 

2 

5 

4 

1 

5 

5 

0 

6 

6 

-  1 

6 

-  1 

6 

5 

-2 

7 

5 

Tabulating  the  values  of    x  —  y  =  3, 
we  have  the  following : 


(2) 


X 

y 

x  —  y 

1 

-2 

3 

2 

-1 

3 

3 

0 

3 

4 

1 

3 

6 

3 

3 

0 

-3 

3 

-  1 

-4 

3 

-2 

-5 

3 

-3 

6 

3 

This  equation  is  indeterminate.  It  is  seen, 
however,  that  the  set  of  values,  x  =  4,  y  =  I, 
occurs  in  both  tables.  That  is,  x  =  4,  y  =  1  will 
satisfy  both  equations.  Thus  the  two  equations 
considered  together  become  a  determinate  system, 
since  they  determine  a  definite  set  of  values  of  x 
and  y  ;  that  is,  x  =  4,  y  =  1.  The  two  equations, 
however,  are  each  satisfied  by  sets  of  values  of  the 
unknown  numbers  which  do  not  satisfy  the  other. 
They  are  therefore  independent  equations. 


250 


Linear  Simultaneous  Equations  251 

362.  Independent  Equations.  Two  or  more  equations  contain- 
ing two  or  more  unknown  numbers,  and  expressing  different 
relations  between  the  unknowns,  are  independent  equations. 

Thus,  x  +  y  =  5  and  x  —  y  =  3  are  independent.  (Why  ?)  x  —  y  =  3 
and  2  x  —  2  y  =  6  are  not  independent  since  the  second  can  be  reduced  to 
the  first  by  dividing  both  members  by  2.  Any  solution  of  one  is  a  solu- 
tion of  the  other. 

363.  Simultaneous  Equations.  Two  or  more  independent 
equations  containing  two  or  more  unknowns  which  are  satisfied 
by  the  same  set  of  values  of  the  unknowns  are  simultaneous 
equations. 

Thus,  x  +  y  =  5  and  x  —  y  =  3  are  simultaneous  equations.     (Why  ?) 

364.  Principles  used  in  Solving  Simultaneous  Equations. 

(a)  If  equal  numbers  are  added  to  equal  numbers,  the  resulting  num- 
bers are  equal. 

(6)  If  equal  numbers  are  subtracted  from  equal  numbers,  the  resulting 
numbers  are  equal. 

Illustration.     3  +  5  =  7+1  (1) 

2  +  3=5  (2) 

3  +  5  +  2  +  3  =  7+  1  +  5.      (Adding  equations  (1)  and  (2) .) 

Let  the  student  subtract  (2)  from  (1)  and  note  the  result. 

365.  Tabulating  values  to  find  a  set  common  to  a  system  of 
simultaneous  equations  is  too  long  a  process.  The  following 
method  is  much  shorter. 

Solve  the  system  of  equations  x  +  y  =  5, 

x-y  =  3. 

Solution.  2x  =  8.     (Adding  the  equations.) 

x  =  4.     (Why?) 
Substituting  this  value  of  x  in  the  first  equation,  we  have 
4  +y  =  5 

y  =  1.     (Why  ?) 

By  adding  the  equations  we  get  rid  of  one  of  the  unknowns ; 
this  process  is  known  as  elimination. 


252  Linear  Simultaneous  Equations 

366.  Elimination.  The  process  of  combining  a  system 
of  equations  so  that  one  of  the  unknown  numbers  disap- 
pears is  called  elimination. 


ORAL  EXERCISE 

367.   Eliminate  x  in  the  following  exercise  and  solve  for  the 
remaining  letter : 


1. 

x  +  y  =  10, 

5. 

M-  *  =  9, 

9. 

2x  +  y  =  7, 

x  —  3y  =  2. 

x  =  7. 

2x  =  0. 

2. 

2x  +  y  =  5, 

6. 

x-z  =  3, 

10. 

2a;  +  3  #=10, 

2x-y  =  l. 

x-2z  =  l. 

32/-2a=-4. 

3. 

y-x  =  7, 

7. 

x  —  w  —  5, 

11. 

£  +  3a  =  4, 

y  +  x  =  9. 

x  -f-  w  =  4. 

t-3x  =  -2. 

4. 

x-2y  =  3, 

8. 

x  -  y  =  a, 

12. 

2x-2a  =  $, 

y  -  x  =  5. 

x  +  y  =  b. 

2x-a  =  S. 

368.  Without  attempting  a  complete  discussion  it  may  be 
stated  that,  in  order  to  solve  a  system  of  linear  equations  with 
two  or  more  unknowns,  three  conditions  are  necessary : 

1.  There  must  be  as  many  equations  as  there  are  unknown  numbers. 

2.  The  equations  must  be  simultaneous  ;  that  is,  there  must  be  a  set 
of  values  of  the  unknowns  that  will  satisfy  all  the  equations. 

8.  The  equations  must  be  independent ;  that  is,  they  must  express 
different  relations  between  the  unknown  numbers. 


369.1   The  preceding  discussion  and  definitions  may  be  made 
clear  by  the  use  of  the  graphs  of  the  equations. 

1  Section  369  may  be  omitted,  if  desired. 


Linear  Simultaneous  Equations 


253 


V             ~ 

5  S/D 

_  2        ^L    nz 

^7 

^^ 

x'                                    /     V 

-5-4-3-2-1           0       1        2     /S         4         g\ 

-l         z 

2 

=2       _^<L 

=3iZ 

2 

7_4 

2 

Z           =5 

Z          F 

Fig.  1 


Tabulating  values  for  the  equations  a:  +?/  =  5  and  x  —  y  =  3, 
we  obtain  the  following : 

If  we  locate  the  points  referred  to 
the  same  set  of  axes  and  draw  the 
lines  through  them,  we  get  two  inter- 
secting lines. 

The  line    (1)    represents    graphi- 
cally equation    (1)    and  the  coordi- 
nates   of    every  point    on    line    (1) 
satisfy  equation   (1).     Line  (2)  rep- 
resents   equation    (2)    and    the    co- 
ordinates   of    every    point    on    line 
(2)  satisfy  equation  (2) .     The  values 
of  x  and   y   at   the   intersection   of 
(1)    and     (2\    or    the    coordinates 
of    the    point    of    intersection    of    the    two    lines,    must    satisfy    both 
equations.     Therefore  x  =  4,    y  .-.  1   is  the  solution  of  the  two  equa- 
tions. 


x+y  = 

:6      (1)             X 

^y  = 

3     (2) 

x 

y 

X 

y 

0 

5 

0 

-3 

1 

4 

1 

-2 

2 

3 

2 

-  1 

3 

2 

3 

0 

4 

1 

4 

1 

5 

0 

5 
-  1 

-2 

2 
-4 
—  5 

254 


Linear  Simultaneous  Equations 


Consider  the  equations        x  -f-  y  =  3, 
2x  +  2y  =  $, 


(1) 

(2) 


and  tabulate  values  for  both. 


x-\-y  = 

3     (1) 

2  x  +  2  y 

=  8    (2) 

X 

y 

x 

y 

0 

3 

0 

4 

1 

2 

1 

3 

2 

1 

2 

2 

3 

0 

3 

1 

4 

-  1 

4 

0 

-  1 

4 

5 

-  1 

—  2 

5 

-  1 

5 

If  we  locate  these  points 
and  draw  the  lines  on  the 
same  axes,  we  have  figure  2. 

The  two  lines  which  repre- 
sent equations  (1)  and  (2)  are 
parallel,  and  hence  have  no 
intersection.  Therefore  there 
is  no  set  of  values  of  x  and  y 
that  will  satisfy  both  equations. 
In  other  words,  the  equations 
are  not  simultaneous. 


v 

\ 

\ 

\ 

\ 

(2) 

0) 

A" 

X 

-.' 

\ 

_ 

I 

- 

1 

0 

i 

l 

\ 

j 

j 

-1 

-0 

-a 

T 

Fig.  2 

Figure  1  represents  the  graph  of  a  system  of  two  simultane- 
ous independent  equations  and  shows  their  solution.  Figure  2 
represents  a  pair  of  equations  that  are  independent,  but  are 
not  simultaneous. 


Elimination  by  Addition  and  Subtraction      255 

ELIMINATION   BY   ADDITION   AND    SUBTRACTION 

Examples 

370.    1.   Solve  the  system  of  equations 

2x  +  y  =  7,  (1) 

Sx-y  =  S.  (2) 

6x  =  10.     (Adding  equations  (1)  and  (2).) 
x  =  2. 
4  +  y  =  7.     (Substituting  the  value  of  x  in  (1).) 

y  =  3. 
Check.     2-2  +  3  =  7.     (Substituting  2  for  x  and  3  for  y  in  both  equa- 
3-2-3  =  3.        tions.) 

2.   Solve  the  system 

2x  +  y  =  25,  (1) 

3a; -2y  =  6.  (2) 

4  x+  2  y =50.     (Multiplying  equation  (1)  by  2.  Why  ?)  (3) 
7x=56.     (Adding  equations  (2)  and  (3).) 
*  =  8. 
16  +  y  =  25.     (Substituting  in  (1) .) 
y  =  9. 
Check.     As  in  example  1. 


3.   Solve  the  system 

[2x  +  ly  =  l, 

(i) 

8z  +  92/  =  4. 

(2) 

24z  +  14  y  =  14. 

(Multiplying  equation  (1)  by  2.)   '                     (3) 

24x  +  27y  =  12. 

(Multiplying  equation  (2)  by  3.)                         (4) 

-13y  =  2. 

(Subtracting  equation  (4)  from  equation  (3).) 

*=-*• 

8*-}f  =  4. 

(Substituting  in  equation  (2).) 

8*  =  H- 

*  =  «. 

Check.               12 • 

M  +  7.(-^)  =  W-^  =  H  =  7. 

8-tt  +  9.(-A)  =  tt-i*  =  H  =  4. 


256  Linear  Simultaneous  Equations 

371.  The  examples  of  the  last  article  should  be  carefully 
studied.  They  illustrate  the  method  of  solving  a  system  of 
simultaneous  equations  by  addition  and  subtraction.  A  de- 
scription of  these  solutions  gives  us  the  rule. 

To  solve  a  system  of  linear  simultaneous  equations  with  two  unknown 
numbers  : 

1.  Multiply  one  or  both  equations  by  such  numbers  as  will  make  the 
coefficients  of  one  of  the  unknowns  the  same  in  both  equations. 

2.  Add  or  subtract  the  resulting  equations  to  eliminate  that  one  of 
the  unknowns  whose  coefficients  are  numerically  equal. 

3.  Solve  the  resulting  linear  equation. 

4.  Substitute  the  value  of  the  unknown  already  found  in  one  of  the 
original  equations,  and  solve  the  resulting  equation  for  the  other 
unknown. 

372.  Before  applying  the  rule  the  equations  are  usually  put 
into  the  form  ax  -f  by  =  c ;  that  is,  the  term  containing  x  is 
written  first,  followed  by  the  term  containing  y,  and  the  known 
term  or  terms  are  in  the  second  member  of  the  equation. 

In  determining  which  unknown  to  eliminate,  study  the  equa- 
tions with  reference  to  the  coefficients  of  the  unknown. 

EXERCISE 

373.  Solve  the  following  systems  : 

1.  3x-2y  =  ±,  6.   2x  +  3y  =  31, 

x  +  2y  =  ±.  3x-y  =  8. 

2.  2x-3y  =  -l,  7.   4r  +  5s  =  40, 
#  +  41/  =  16.  6r-7s  =  2. 

3.  2t-3u  =  -l,  8.   2av+#2  =  7, 

t  —  u  =  l.  —  2  xx  +  3  x2  =  13. 

4.  a;  -h  5  2/  =  3,  9.   7  x'  -  3  x"  =  15, 
4x-2y  =  l.  5  a' -6 a"  =  27. 

5.  %m  +  %n  =  %y  10.   8  a  +  17  b  =  42, 
m-n  =  \.  2  a  +  19  b  =  40. 


Elimination  by  Addition  and  Subtraction      257 

11.  28ra  +  rc  =  33,  16.   2x—  11  y  =  —  95, 
-  21  m  +  11  w  =  34.  x  -  3  y  =  0. 

12.  12r1-5r2  =  64,  17.    12  a; -5  #  =  24, 
8  rx  +  3  r2  =  68.  3  x  +  10  y  =  6. 

13.  7  x- 12 y  =115,  18.    7v  —  15w  =  -45, 
2  a  +  5 1/  =  16.  8  v  +  5 10  =  15. 


14. 

8  x  +  3  ?/  =  37, 

19.   4:X  —  3y  =  l, 

8  y  -  3  x  =  50. 

7x  =  3.5. 

15. 

8  s  +  21  *  =  649, 

20.   £  x  +  3  y  =  25, 

14  s  -  9 1  =  541. 

8a;  +  i#  =  65. 

21.    M  = 

»    y 

=  10,                                     (1) 

a;     2/ 

=  20.                                     (2) 

Solution.     -  +  -  =  40. 
x     y 

(From  equation  (1).)                                   (3) 

5  =  20. 
y 

(Subtracting  equation  (2)  from  equation  (3).) 

20  y  =  5. 

(Why?) 
(Why?) 

-  +  8  =  10. 
a; 

1  =  2. 

X 

(Substituting  in  equation  (1).) 

(Why  ?) 

22. 

1,1     5 
a;     2/     o 

24.    -1-.J—1, 

2a;     3y     4' 

1_1_1 
x     y     6 

3a?     42/     2 

23. 

M  =  3, 
a;     y 

25.    -~82/  =  L 
a?                b 

15_4_4 

9     y 

3i0         2 
a;               15 

258  Linear  Simultaneous  Equations 


Solve  the  following  systems  : 

26.   2x-3y  = 

=  5  b  —  a, 

(1) 

3x-2y= 

=  a  +  5  6. 

(2) 

Solution.          6x  —  9y  =  15  b  —  8  a. 

(Eq.  (1)  x  3.) 

(3) 

6x-4y  =  10b  +  2a. 
—  5y  =  5b  —  5a. 
y  =  a—b. 
2x  —  3(a  —  b)=5b-a. 
2x-Sa  +  3b  =  6b-a. 
2x  =  2a  +  2b. 
x  —  a  +  b. 

(Eq.  (2)  x  2.) 

(Eq.  (3)  -  Eq.  (4).) 

(Why  ?) 

(Why  ?) 

(Why  ?) 

(Why  ?) 

(Why  ?) 

(4) 

27.   2  x  +  y  =  2  a, 

35.    7x-3y  =  27, 

2  x  -  y  =  2  6. 

x  :  y  =  6  :  5. 

28.   5a-22/  =  5a-2  6, 

36-    J**-* 

29.   x  +  ^  =  i(5aH-&), 

•*. i/  —  1  /"•»  _i_  K  M 

£    '            —      3* 

6      ?/ 

30.  2x+3y=10a-2b+3c,  37'   ™  +  t>y  =  2a, 
x-2yL-2a-b-2c.  a*x-Vy  =  a>  +  V. 

31.  5aj  +  3y  =  4a  +  6,  38«   a>4-w#  =  -l, 
3*+5y  =  4a-6.  y  «■  n(*  + 1). 

32.  a  +  2/  =  10  a  -3  6,  39.    a  +  1  =  a?/, 
2  a  —  y  =  2  a  +  3  b.  y-bx  =  b. 


33.   az+&2/  =  a,  4Q    3  a*  + &2/ =  a2  +  1, 

?*-y=2 
6        y      b 


*x-y  =  -.  (>x-2b*y  =  2a-2b. 


34.   10 +  7  y  4- 4=0,  41.   3a  +  2  y  =  8  a-  7  6, 

6  a  4-  5  2/  4-  2  =  0.  ax  +  by  =  2a*-2b2. 


Elimination  by  Substitution 


259 


ELIMINATION   BY   SUBSTITUTION 

374.  Principle  of  Substitution.     Any  number  may  be  substi- 
tuted for  its  equal. 

375.  1.     Solve  the  system  2  x  +  y  =  25,  (1) 

3x-2y  =  6.  (2) 

Solution.  y  =  25  —  2x.     (From  equation  (1).)    (3) 

Sx  -  2(25  -  2x)  =  6.  (Substituting  in  (2).) 

3x-60  +  4x  =  6. 
7s  =  56. 

85=8. 

y  =  25  -  2  •  8  =  9.     (From  (3).) 
Compare  this  with  example  2,  §  370. 

2.    Solve  the  system     x  +  y=  15,  (1) 

x:y  =  2:Z.  (2) 

Solution.  3x  =  2y.     (From  (2).)     (Why?)      (3) 

x  =  \y.  (4) 

|  y  +  y  =  15.      (Substituting  in  (1) .) 
|  y  =15. 
2/ =9. 

x  =  6.       (Substituting  in  (4).) 
Let  the  student  check  mentally. 


3.    Solve  the  system  3  x  —  4  y  =  8, 


Solution. 


(1) 
(2) 


*=§J3^'     (^"WO  (3) 


4/8+^jA  +  3  y  =_  6  (Substituting  in  (2).) 


Check  mentally. 


32  +  16^  +  Sy=-6. 

32+  16y  +  9y  =  —  18. 

25  ?/  =  -  50. 

y=-2. 

=  8+4.(-2)=Q 
3 


(From  3).) 


260  Linear  Simultaneous  Equations 

376.  To  solve  a  system  of  simultaneous  equations  by  substitution : 

1.  Find  the  value  of  either  of  the  unknowns  in  terms  of  the  other 
unknown  and  known  numbers  from  one  of  the  equations. 

2.  Substitute  the  value  of  this  unknown  for  the  same  unknown  in  the 
other  equation. 

3.  Solve  the  resulting  equation. 

4.  Substitute  the  value  of  the  unknown  that  has  been  found  in  one  of 
the  preceding  equations  to  find  the  other  unknown. 

377.  The  method  of  solving  simultaneous  equations  by 
substitution  is  especially  convenient  when  one  of  the  unknowns 
can  readily  be  expressed  in  terms  of  the  other.  It  is  also 
much  used  in  later  work  and  should  be  well  understood. 

EXERCISE 

378.  Solve  the  following  systems  by  the  method  of  substitution : 

1.  2x-lly  =  -95,  9.    7x  -  y  -  6a  =  126, 
x  —  3  y  =  0.  x  =  y. 

2.  B*-2yW21,  l0    ax  +  by  =  c, 


x :  y  =  5  : 2. 
3.   fc-2d  =  l, 


x  =  2y. 


ie-d  =  0.  «'   ^-2^  =  69, 


4.   x  =  3y  -19, 


2x  +  y  =  78. 


y  =  3x-23.  15      M 

*  12.   m  +  —  =  13, 


5.   2w-%y  =  4=, 


n 


3w-$y  =  0.  m  +  ^  =  16. 


n 


6.  5u-  4.9v  =  l, 

3w-2.9<y  =  l.  13.   x-f(?/  +  l)=3, 

7.  3x  +  162/  =  5,  K*-l)-i2/  =  *i. 
28?/ -5a;  =  19. 


7-2^3 

5-32/     : 
+  t  =  \.  y-x  =  4:. 


8.   f«  +  *  =  f,  '    5-3y     2' 


Linear  Simultaneous  Equations  261 

21     x+m=p 


15. 

x  —  6      2 
y  +  2     31 

x  +  1     3 

y-2     2 

16. 

a?  +  4:y  +  l 

=  2 

:1, 

s  +  2:y-I 

=  3 

:1. 

17. 

«#  +  ?/  =  m, 

x  —  y  =  n. 

18. 

x  +  my  =  a, 
x  —  ny  =  b. 

19. 

x  :  y  =  a  :  b, 

•  +l:jf+l 

=  c: 

d 

20. 

771    .    1 

-  4-  -  =  p, 
a      2/ 
n  .  1 

23       ?/ 

22. 


23. 


2/ 

—  71 

-    ) 

q 

qx+py 

=  s. 

X 

m 

+  y-  = 

n 

=p> 

r 

=  ?. 

X 

2/ 

t- 

-3 

p 

u         qf 

?  =  ?. 

t     u 


24.    -  +  -  =  m, 
x     y 

l_l=ri 
»     2/ 


379.  Most  of  the  equations  thus  far  solved  have  been  given 
in  the  form  ax  -f-  by  =  c.  In  the  following  exercises  the  equa- 
tions should  be  simplified  before  applying  the  rule. 

EXERCISE 

380.  Solve  by  either  method  : 

1.   4(3a>-5)-2(y-3)=2> 
2(5x-y)-3y=5. 

2  7       -      1 

Ax  —  y      x  —  y 

2(x-y)=y-S. 
3.   4(*-3y)=8, 


262  Linear  Simultaneous  Equations 

Solve  by  either  method : 

4.  x+y  =  lO, 

3x  +  5y     5x  +  3y_x  +  5y     5 
4~  11  11      *2* 

5.  5x-(3y-i)=.75, 
4  +  oj-2(2/-1)=0. 

6.  a:  (b  +  y)=b:(3a  +  x)} 
ax  +  2  by  =  b2. 

7.  x:y  =  3:4, 
x-l:y  +  2  =  l:2. 

8.  icH-l:i/  +  l:a;  +  2/  =  3:4:5. 

9.  x  —  5:y  +  9:x  +  y  +  9=:l:2:8. 

10        2a?+y  — 1     =1     5  a;  — 3y  +  4_3t 
3a  +  22/  +  ll~2'    6a-3?/-r-3~~4' 

11.  bcx  =  cy-2b,   6V  +  a(c3~63)  =  —  +  eta. 

6c  c 

12.  (a-6)x+(a  +  %  =  a  +  J 


a  +  6     a  —  6     a-f-6 


13.    "±!-«i    £±±=&. 


X 


a2  —  62  a2  —  62 

(Do  not  clear  of  fractions.) 

16.    (a  +  6)*  +  (a  -  6)y  =  2(a2  +  V), 
(a  -  6)*  +  (a  +  b)y  =  2(a2  -  62). 
(First  add  the  equations,  then  subtract  them.) 


Linear  Simultaneous  Equations  263 

17.  s  +  y  +  l  =  a  +1     a?  +  y  +  l  =  6-t-lT 
a  —  y  +  1      a  — 1'    a;  — I/  — 1      6  —  1* 

(Use  composition  and  division  on  both  equations.) 

18.  (z-f5)(y  +  7)  =  (a;  +  l)(y-9)+112, 
2a;  +  10  =  3y  +  l. 

19.  2.60  a;  -  .41  y  -  4.28  +  2.50  x  =  0, 
.50  a;  +  3.6  y  +  3.23  +  .5  y  =  11.93. 

20     g  +  1-.y+g-20g-y) 
3  4  ~5       ' 

a;-3      y-3      rt 


21     3  a;  —  2  y     5  a;  —  3  y 


=  *  +  l, 


2a,--3y.4a;-3y 

22.    4(a._2/)__iffa,__iT_2/=14) 
f(z-14)-^(y  +  12)  =  -2. 

23     10a;-2y  +  22      7a?-3y  =  a?  +  y-l 
56  14  8 

9         i8     -7*-^- 

24.  2(a;  +  y-c)=2(a;-c)+a;  +  3y-c, 
a;  +  7  y  =  15  c. 

25.  (»-l)(5y-3)=s3(3a?  +  l)+5ay, 

(s-l)(4y  +  3)=3(7y-l)+4a>y. 

26.  -4-+-JL.  =  «  +  &, 

a  +  b     a  —  b 

X    I    V        o 

a     6 


264  Linear  Simultaneous  Equations 

Solve  by  either  method : 

27.    (a  -f-  c)x  —  (a  —  c)y=2  ab, 
(a  +  b)y  —  (a  —  b)x  =  2ac. 
(Add  the  equations  and  simplify.) 

28.  5x-y=Sx-5y=y-3. 

11  4  * 

(Put  each  of  the  first  two  equal  to  the  last  expression.) 


29.    ?  =  !«»  +  ?  +  <?. 

a     b 

o0        7-6r_4-3r 
10* -19     5*- 11' 

32.    .75  a;  +  .8  y  =  21, 
x  _y 

6  r  -  10  *  -  17     4  r  - 

- 14  f  -  5 

4     5 

3r_5^  +  2       2r- 

-  7  *  +  12 

31.     -:  -  =  13:11, 
c    d 

33.    5  +  §=3, 
x     y 

5c  +  3      7d-4 
2               3 

1^-4=4. 

PROBLEMS  WITH  TWO  UNKNOWN  NUMBERS 

381.  Many  of  the  problems  previously  solved  with  one  un- 
known number  could  have  been  solved  with  two  unknowns. 
The  problems  following  are  to  be  solved  using  two  unknown 
numbers. 

1.    Find  two  numbers  whose  sum  is  40  and  whose  difference 

is  22. 

Solution.     Let  x  =  the  greater  number, 

and  y  =  the  smaller  number. 

Then  x  +  y  =  40, 

and  x  —  y  =  22. 

2  x  =  62. 

3  =  31. 

y  =  9. 


Problems  with  Two  Unknown  Numbers       265 

2.  The  sum  of  two  numbers  is  21,  and  two  times  the  first 
exceeds  3  times  the  second  by  2.     Find  the  numbers. 

3.  Find  two  numbers  whose  sum  equals  95  and  whose 
ratio  is  2  :  3. 

4.  Two  numbers  are  in  the  ratio  of  4  to  7.  .If  four  is 
added  to  each  of  the  numbers,  the  results  are  in  the  ratio  of 
£  to  3.     Find  the  numbers. 

5.  The  sum  of  the  reciprocals  of  two  numbers  is  ^ ,  and 
the  ratio  of  the  numbers  is  i.     Find  the  numbers. 

6.  If  the  first  of  two  numbers  is  multiplied  by  3  and  the 
second  by  8,  the  sum  of  the  products  is  310 ;  if  the  first  is 
divided  by  3  and  the  second  by  8,  the  sum  of  the  quotients  is 
10.     What  are  the  numbers  ? 

7.  The  sum  of  two  numbers  is  350.  If  the  first  is  divided 
by  the  second,  the  quotient  is  8  and  the  remainder  is  8.  What 
are  the  numbers  ? 

8.  Find  two  numbers  whose  difference  and  quotient  are 
each  equal  to  10. 

9.  Solve  the  last  problem  using  a  in  place  of  10. 

10.  A  and  B  together  have  $  120.  If  A  would  give  B  one 
third  of  his  money,  they  would  have  equal  amounts.  How 
much  has  each  ? 

Solution.     Let  x  =  number  of  dollars  A  has, 
and  y  =  number  of  dollars  B  has. 
Then  x  +  y  =  120.     (By  the  first  condition.) 
Also  |  x  =  number  of  dollars  A  has  after  giving  B  |  of  his  money, 
and  y  +  \x  =  number  of  dollars  B  lias  after  receiving  \  of  A's  money. 

Then  |x  =  y  +  \x.     (By  the  second  condition.) 
Solve  the  equations  and  verify  the  answers  by  putting  them  in  the 
original  problem. 

11.  A  and  B  have  a  certain  amount  of  money.  If  A  had 
$  15  more,  he  would  have  as  much  as  B.  If  B  had  %  15  more, 
he  would  have  twice  as  much  as  A.     How  much  has  each  ? 


266  Linear  Simultaneous  Equations 

12.  If  B  gives  A  $  5  they  will  have  equal  amounts  of  money ; 
but  if  A  gives  B  $  5,  B  will  have  twice  as  much  as  A.  How 
much  has  each  ? 

13.  A  resolution  was  carried  by  a  plurality  of  20  votes. 
On  reconsideration  \  of  those  voting  for  it  changed  their  votes 
and  it  was  lost  by  12  votes.  How  many  voted  each  way  tht 
first  time  ?  • 

14.  A  dealer  has  two  kinds  of  coffee,  worth  30  cents  and  40 

cents  a  pound  respectively.     How  many  pounds  of  each  must 

he  take  to  make  a  mixture  of  70  pounds,  worth  36  cents  a 

pound  ? 

Solution.     Let  x  =  number  of  pounds  30-cent  coffee, 

and  y  =  number  of  pounds  40-cent  coffee. 

Then  x  +  y  =  70, 

and  .30 x  +  AOy  =  25.20.     (70  x  $  .36  =  $25.20.) 

Let  the  student  explain  these  equations  and  solve  them. 

15.  A  dozen  oranges  and  6  pineapples  cost  $1.30.  Six 
oranges  and  2  pineapples  cost  $  .50.  Find  the  cost  of  an  orange 
and  of  a  pineapple. 

16.  A  ruble  is  a  Russian  coin  and  a  mark  is  a  German  coin. 
Two  rubles  and  3  marks  are  worth  $  1.75  in  our  money ;  also 
a  ruble  is  worth  3|  $  more  than  two  marks.  Find  the  value  of 
each  in  our  money. 

17.  A  man  invested  $  5000,  a  part  at  5  %  and  the  remainder 
at  6  <f0 .  The  interest  amounted  to  $  265  annually.  How 
much  was  on  interest  at  each  rate  ? 

Suggestion.  One  equation  is  .05  x  +  .06  y  =  266.  Let  the  student 
make  the  other. 

18.  69  quarters  and  dimes  are  worth  $9.45.  How  many 
of  each  are  there  ? 

19.  40  stamps,  some  one-cent  and  the  rest  two-cent  stamps, 
cost  65  $.     How  many  of  each  were  bought  ? 


Problems  with  Two  Unknown  Numbers       267 

20.  The  units'  digit  of  a  number  of  two  figures  exceeds  the 
tens'  digit  by  1 ;  the  number  divided  by  the  sum  of  its  digits 
is  equal  to  5.     Find  the  number. 

Solution.  Let  x  =  the  tens'  digit, 

and  y  =  the  units'  digit. 
Then  x  +  1  =  y.     (Why  ?) 
Also  10  x  +  y  =  the  number.     (Why  ?) 

Then10x  +  y  =  5. 
x  +  y 
10x  -\-  y  =  bx  +  5y. 
.-.  bx  —  ±y  =  0. 
.-.  5z-4(z  +  l)=0. 
.-.  a  =  4, 
and  y  =  x  +  1  =  5- 
Therefore  the  number  is  45. 

21.  If  the  digits  of  a  number  of  two  figures  are  inter- 
changed, the  number  obtained  is  -|  of  the  original  number. 
The  units'  digit  exceeds  the  tens'  digit  by  5.     Find  the  number. 

22.  If  the  digits  of  a  two-figure  number  are  interchanged, 
the  resulting  number  multiplied  by  2  exceeds  the  original 
number  by  1.  If  the  number  is  divided  by  the  sum  of  its 
digits,  the  result  is  7.3.     Find  the  number. 

23.  What  two-figure  number,  the  sum  of  whose  digits  is  10, 
has  the  property  that  if  its  digits  are  interchanged,  the  num- 
ber is  diminished  by  36  ? 

24.  The  value  of  a  fraction  is  -|.  If  the  numerator  and  the 
denominator  are  both  diminished  by  18,  the  value  is  ^.  What 
is  the  fraction  ? 

Solution.  Let  n  =  the  numerator, 

and  d  =  the  denominator. 

Then  ^  =  2 
d     3 

and!^i8=i. 
d-18      3 

Let  the  student  solve.     The  answer  is  f  f . 


268  Linear  Simultaneous  Equations 

25.  The  value  of  a  ratio  is  5.  If  5  is  subtracted  from  the 
antecedent  and  5  is  added  to  the  consequent,  the  value  is  2\. 
Find  the  antecedent  and  the  consequent. 

26.  A  sum  of  money  was  divided  equally  among  a  number 
of  people.  Had  there  been  3  more  people  each  would  have 
received  $  1  less.  Had  there  been  5  less  each  would  have  re- 
ceived $  3  more.  How  many  people  were  there  and  how  much 
did  each  receive  ? 

Solution.  Let  x  =  number  of  people. 

and  y  =  number  of  dollars  each  received. 
Hence  xy  =  number  of  dollars  divided. 
Then  (x  +  3)(y  -  1)=  xy, 
and  (x  -  5)  (y  +  3)  =  xy. 
Let  the  student  solve  these  equations.     The  number  of  people  was  15. 

27.  A  and  B  can  do  a  piece  of  work  in  4|  days.  After  A 
and  B  work  together  for  2  days,  B  can  finish  the  work  in  7 
days.     How  long  will  it  take  each  ? 

Suggestion.  If  A  can  do  the  work  in  x  days,  what  part  of  it  can  he 
do  in  one  day  ? 

28.  It  is  known  that  gold  loses  -^  of  its  weight  when 
weighed  in  water,  and  silver  loses  y1^  of  its  weight.  The  gold 
and  silver  crown  of  King  Hiero  of  Syracuse  weighed  20  pounds 
and  lost  1 J  pounds  in  water.  How  much  of  it  was  gold  and 
how  much  was  silver  ? 

29.  Two  boys  run  on  a  circular  track  which  is  90  yards 
around.  If  they  run  in  opposite  directions,  starting  at  the 
same  time,  they  meet  in  5  seconds  ;  but  if  they  run  in  the  same 
direction,  the  faster  will  overtake  the  slower  in  45  seconds. 
How  many  yards  a  second  can  each  boy  run  ?  How  long 
would  it  take  each  to  run  100  yards?  At  what  point  of  the 
track  do  they  meet  when  they  run  in  opposite  directions  ? 
At  what  point  are  they  together  when  they  run  in  same 
direction  ? 

Hint.    The  equations  are  5x+5*/=90  and  45x-45y  =  90.     (Explain.) 


Problems  with  Two  Unknown  Numbers       269 

30.  A  and  B  have  $  120  between  them.  If  A  spends  one 
third  of  his  money  and  B  spends  $  10,  they  will  have  only 
$  85.     How  much  has  each  ? 

31.  The  area  of  a  rectangle  is  unchanged  if  the  length  is 
diminished  by  4  inches  and  the  width  is  increased  by  4  inches. 
It  is  increased  by  40  square  inches  if  both  dimensions  are 
increased  by  2  inches.     Find  the  length  and  the  breadth. 

32.  A  man  walks  3  miles  an  hour  up  hill  and  4i  miles  an 
hour  down  hill.  In  walking  from  A  to  B  on  a  road  no  part  of 
which  is  level,  he  requires  6^  hours ;  but  to  walk  from  B  to  A 
he  requires  only  6  hours.  How  much  of  the  road  from  A  to 
B  is  up  hill  and  how  much  is  down  hill  ? 

33.  In  making  the  run  between  two  ports  a  boat  averages 
14  miles  an  hour.  On  a  certain  trip  it  runs  at  its  usual  rate 
for  5  hours  and  then,  on  account  of  a  fog,  is  obliged  to  proceed 
at  one  half  of  its  regular  speed,  arriving  in  port  4  hours  late. 
What  is  the  distance  between  the  two  ports  ? 

34.  The  circumference  of  a  circle  contains  360°.  Find  the 
number  of  degrees  in  each  of  two  arcs  into  which  it  is  divided 
if  their  difference  is  240°. 

35.  Divide  $  10,000  between  two  persons  so  that  one  of  them 
shall  receive  ^-  as  much  as  the  other. 

36.  A  farmer  bought  10  cows  and  sold  16  sheep,  having  to 
pay  out  $  446  in  excess  of  what  he  received.  The  next  day 
he  bought  3  cows  and  sold  12  sheep  at  the  same  price,  paying 
the  difference  of  $  72.     Find  the  cost  of  one  cow  and  one  sheep. 

37.  The  expression  ax2  +  foe  —  30  is  equal  to  330  when 
x  =  5  and  is  equal  to  64  when  x  =  12.  Find  the  values  of  a 
and  b. 


38.   A  fraction  is  equal  to  f  when  10  is  added  to  its  numer- 
,or,  and   to  i  when  4  is  sub 
Find  the  value  of  the  fraction. 


ator,  and   to  i  when  4  is  subtracted    from   its   denominator. 


270  Linear  Simultaneous  Equations 

39.  The  numerator  of  a  fraction  is  4  less  than  the  denomi- 
nator. If  16  is  subtracted  from  the  numerator,  or  if  36  is 
added  to  the  denominator,  the  resulting  fractions  will  be 
equal.     Find  the  value  of  the  fraction. 

40.  The  law  of  a  machine  is  given  by  E  =  aR  +  b,  when 
E  =  efficiency  of  the  machine  and  R  =  resistance  due  to  fric- 
tion. When  E  =  4.2,  R  =  10,  and  when  E  =  7.34,  R  =  20. 
Find  the  values  of  a  and  b. 

41.  The  expression  mx  —  y+b  is  equal  to  22,  when  x  =  5 
and  y  =  2.  It  is  equal  to  12  when  x  =  —  3  and  y  =  —  4.  Find 
the  values  of  m  and  b. 

42.  The  expression  -  +  \  —  1  is  equal  to  5  when  x  =  2  and 

a     b 

y=3.     It  is  equal  to  4  when  x  =  3  and  y  =  2.     Find  the  values 
of  a  and  b. 

43.  If  a  :  y  =  11 :  15,  find  what  values  of  x  and  y  will  satisfy 
the  equation  5  x  +  7  ?/  =  32. 

44.  The  sum  of  the  angles  of  a  triangle  is  180°.  How  many 
degrees  are  there  in  each  of  the  acute  angles  of  a  right-angled 
triangle,  if  one  of  the  acute  angles  is  three  times  the  other  ? 

45.  if  3a?  +  4^  =  3,  find  the  ratio  of  x  to  y. 

4:X  —  Sy 

46.  What  value  of  x  and  y  will  make  the  two  expressions 
2x  +  5  y  —  23  and  Ax  —  Sy  each  have  the  value  zero  ? 

47.  The  price  of  admission  to  a  moving  picture  show  is  5^ 
for  children  and  10  ^  for  adults.  435  tickets  are  sold  and  the 
receipts  are  $  30.  How  many  children  and  how  many  adults 
attend  the  show  ? 

48.  Brass  is  an  alloy  of  copper  and  zinc.  If  copper  is  16  p 
a  pound  and  zinc  is  5tf  a  pound,  how  many  pounds  of  each 
must  be  used  to  make  300  pounds  of  brass  that  is  worth 

$37?. 


Three  Unknown  Numbers  271 

49.  Bell  metal  is  an  alloy  of  copper  and  tin.  The  value  of 
the  material  in  a  bell  weighing  400  pounds  is  $82.56.  If 
copper  is  18^  a  pound  and  tin  is  30^  a  pound,  how  many 
pounds  of  each  metal  are  there  in  the  bell  ? 

50.  Nickel-steel  for  automobile  construction  is  an  alloy 
of  steel  and  nickel.  The  value  of  the  material  in  3825 
pounds  is  $  74.50.  If  steel  is  1  ^  a  pound  and  nickel  is  30  ^ 
a  pound,  how  many  pounds  of  each  metal  are  there  in  the 
3825  pounds? 

THREE  UNKNOWN  NUMBERS 

Examples 

382.    1.   Solve  the  system  x  +  5  y  +  6  z  =  29,  (1) 

10z  +  2/  +  2z  =  18,  (2) 

5z  +  9y  +  3z  =  32.  (3) 

30 £  +  3 y  +  6  z  =  54.               (Multiplying  equation  (2)  by  3. )  (4) 
—  29x  +  2y=— 25.           (Subtracting     equation     (4)  from  equa- 
tion (1).)  (5) 
10  x  +  18  y  +  6  z  -  64.                (Multiplying  equation  (3)  by  2.)  (6) 
9x  + 13^  =  35.               (Subtracting    equation     (1)    from  equa- 
tion (6).)  (7) 
-  377  x  +  26  y  =  -  325.         (Multiplying  equation  (5)  by  13.)  (8) 
18  x  -f  26  y  =  70.               (Multiplying  equation  (7)  by  2.)  (9) 
—  395x  =  — 395.         (Subtracting    equation    (9)    from  equa- 
tion (8).) 
*sl. 

y  =  2.  (Substituting  x=l  in  equation  (5)  or  (7).) 

z  =  3.  (Substituting    x  =  l,   y  =  2    in    equations 

(1),  (2),  or  (3).) 

Check.      1  +  5  .  2  +  6  •  3  =  29. 

10.1  +  2  +  2.3  =  18. 

5- 1+9- 2  +  3-3  =  32. 


272  Linear  Simultaneous  Equations 

2.    Solve  the  system  5u  +  3v  +  2w  =  211,  (1) 

5u-3v  =  39,  (2) 

3  v  -  2  w  =  20.  (3) 

6  v  +  2  w  =  178.  (Subtracting  equation  (2)  from  equation  (1).)  (4) 

9v  =  198.  (Adding  equations  (3)  and  (4).) 

v  =  22. 

5m  =  105.  (Substituting  v  =  22  in  (2).) 
w=21. 

io  =  23.  (Substituting  t>  =  22  in  (3).) 
Let  the  student  check  the  results. 

383.  To  solve  a  system  of  linear  simultaneous  equations  with  three 
unknowns : 

1.  Transpose,  if  necessary,  so  that  the  unknowns  will  all  be  on  the 
left-hand  side  of  the  equations  and  collect  like  terms. 

2.  Examine  the  equations  to  see  what  unknown  can  be  most  easily 
eliminated. 

3.  Eliminate  that  unknown,  using  two  of  the  equations. 

4.  Eliminate  the  same  unknown,  using  the  third  equation  and  one  of 
the  others. 

5.  Solve  the  resulting  system  for  the  two  remaining  unknowns. 

6.  Substitute  the  values  of  these  two  unknowns  in  one  of  the  original 
equations  to  find  the  third  unknown. 

384.  In  the  solution  of  a  system  of  simultaneous  equations 
involving  three  unknowns  the  following  suggestions  will  be 
found  useful : 

1.  If  one  of  the  three  equations  contains  but  two  of  the 
three  unknowns,  eliminate  the  third  unknown  from  the  other 
equations.  There  will  then  be  two  equations  containing  two 
unknowns  which  can  be  solved,  according  to  §  371.  (See  ex- 
ample 2,  §  382.) 

2.  In  general,  eliminate  first  the  unknown  that  can  be  most 
easily  eliminated.  If  the  coefficients  of  this  unknown  are  not 
the  same  in  two  of  the  equations,  make  them  the  same  by  using 
the  smallest  multipliers  possible, 


Three  Unknown  Numbers  273 

3.  Addition  or  subtraction  is  usually  the  simplest  method 
of  elimination. 

4.  Study  the  model  solutions  and  the  suggestions  given  for 
special  methods  of  shortening  the  work. 

385.  To  solve  a  system  of  four  equations  with  four  unknowns, 
eliminate  one  of  the  unknowns  by  using  the  four  equations  in  pairs 
three  times,  thus  deriving  three  equations  with  three  unknowns.  Con- 
tinue with  these  equations  as  indicated  in  §  383. 

EXERCISE 

386.  1.   How  can  x  be  eliminated  from  the  system 

x+y  +  z  =  12, 
4sc  +  3y  +  5z  =  49, 
5x-2y  +  z  =  12? 

2.  How  can  y  be  eliminated  from  the  above  system  ?     How 
can  z  be  eliminated  ? 

3.  Which  letter  will  be  most  easily  eliminated  ? 

4.  Which  unknown  should  be  eliminated  first  in  solving 
the  system 

2x-3y  +  7z  =  6, 
Sx  +  4y  +  llz  =  18, 
y-3z  =  -2? 

5.  Solve  the  system  of  equations  given  in  example  1. 

6.  Solve  the  system  of  equations  given  in  example  4. 

7.  Can  you  find  a  definite  solution  of  a  system  of  two  equa- 
tions containing  three  unknowns  ? 

Solve  the  following  : 

8.  2x  +  5y-3z  =  13,  10.  2x  4-  3y  =  12, 
6  x  -  3  y  +  4  z  =  16,  3x  +  2z  =  ll, 
5x  +  3y-6z  =  W.                        3  y  +  4  z  =  10. 

9.  3x-y  +  z  =  7,  11.  2u-7v  =  9, 
x  +  2y  —  4z  =  - 8,  u  +  4:v  =  12, 

2x  —  2y  +  z  =  2.  u  +  v  +  2  w  =  14. 


274 


Linear  Simultaneous  Equations 


Solve  the  following : 

12.  x+2y-.7z  =  21, 
3x  +  .2y-z  =  24:, 
.9x  -\-7y-2z=27. 


21.    X  =  ^- 

7      10 


13. 

lx-iy  =  0, 

\x  —  \z  =  l, 
\z-\y  =  2. 

14.  p  +  q  +  r  =  36, 
4p  =  3?, 
2p  =  3r. 

(Try  substitution.) 

15. 

x  +  y  +  z  =  100, 
y  =  .7x-4, 
z  =  .3  x  4  4. 

16. 

»  +  2/  4  2  =  26, 
x :  %  =  11 :  T, 
2/:z  =  14:9. 

17. 

r  +  »  4-  *  =  99, 
r  :  s  :  *  =  5  :  3  : 1. 

18. 

a?  +  y  +  2  =  «, 
x  _y  _z 
a     b     c 

19. 

x+y  =  c, 
y  +  z  =  a, 
z  4  x  =  b. 

20.   2jc  +  y  +  3  =  a, 

a  +  2?/  +  z  =  a, 

(Add  all  the  equations  and  di- 
vide by  4.) 


z 

5' 
2x+3y  =  88. 


22.  1  +  ^  +  ^+1  =  0, 

3  6      9 

i  +  ^  +  .ZL+1-O 
6+9+12+1~U' 

i+^+iL  +  i=0. 
9^12^15^ 

23.  -  +  -  =  a, 

x     y 

-+-  =  &, 

x     z 

-+-=c. 
2/     ^ 

24.  a  +  b  +  c  =  m, 
&  4-  c  4-  d  as  n, 

c  +  d  4-  a  = '  v, 
d  +  a  4-  6  =  q. 

(First  add  all  equations,   then 
divide  by  3.) 

25.  x  4  y  4  z  4  w;  =  a, 
#  —  2/4z  —  w  =  b, 
x  +  y  —  z  —  w  =  cf 
x  —  y  —  z-\-w  —  d. 

26.  ?-A+2=a^, 

a?     Sy 

_L  +  i  +  2z  =  6H, 

4  a;     ?/ 

6z     y  ** 


Three  Unknown  Numbers  275 

27.  x  +  2y  =  %  31.   x  +  3y=±£> 
3  y  +  4  z  =  14,  a;  -f-  5  ?/  =  3  z, 

7z  +  u  =  5,  10y-3z  +  2  =  0. 

2u  +  5z  =  8. 

28.  a; +  2?/ -z  =  4.6,  32.   3(a  +  2?/)- 2  =  2, 
2/  +  2  z  -  a  =  10.1,  x  +  2  y  =  i  z, 

2  +  2  a  -  J/  =  5.7.  x  +  y  +  2  —  5|. 

29.  3a;  +  22/  +  3z  =  110, 

5a  +  2/  =  4Z,  33    fr^-y-l, 

2-0  +  *-**  a?  +  2g  =  2 

30.  z  =  2/+l,  3  3' 

•-•  +  *  2/--  =  17. 

2/ =  2z -10.  y      6      - 

M     Sx  +  y-z_Sx  +  3y-A  =  2x  +  y_Zf 
3  6 

20  +  y  +  »  —  6. 

35.  a  +  26+3c  =  32,  2a+36  +  c  =  42,  3a  +  6+2c  =  40. 

36.  0  +  y  +  z  =3,  2x  +  4y  +  82  =  13,  3a  +  9#  +  27z  =  34. 

37.  a  +  ?/  +  2 2  =  34,  x  +  2 y  +  2  =  33,  2 a;  +  y  +  2  =  32. 

38.  x=2%y-6,  y  =  3\z-l,z  =  l\x -8. 

PROBLEMS   SOLVED   WITH  THREE   UNKNOWNS 

387.  1.  The  sum  of  three  numbers  is  100.  If  the  first  is 
divided  by  the  second,  the  quotient  is  5  and  the  remainder  1. 
The  second  divided  by  the  third  gives  the  same  result.  What 
are  the  numbers  ? 

2.   Find  three  numbers  whose  sum  is  999,  and  which  are 
to  each  other  as  2  :  3  : 4.     Solve,  using  three  unknowns. 


276  Linear  Simultaneous  Equations 

3.  Solve  problem  2  using  only  one  unknown. 

4.  A  number  equals  the  sum  of  two  other  numbers.  The 
largest  number  diminished  by  2  equals  three  times  the 
smallest.  The  largest  increased  by  2  equals  twice  the  re- 
sult of  diminishing  the  middle  number  by  2.  Find  the 
numbers. 

5.  Find  three  numbers  such  that  if  the  sum  of  each  two  is 
diminished  by  the  other  the  results  are  respectively  0,  4,  and  8. 

6.  Three  men,  A,  B,  and  C,  working  together  can  do  a  piece 

of  work  in  5^  days.     A  and  B  together  can  do  the  work  in  6^ 

days.     A  and  C  can  do  it  in  9f  days.     How  long  will  it  take 

each  one  working  alone  ? 

Suggestion.     One  equation  is  -  -\ \--  = — 

x     y     z     51 

7.  Suppose  they  all  work  together 
and  receive  %  96  for  doing  the  work, 
how  much  should  each  one  receive? 

8.  In  the  figure  the  circles  touch 
each  other.  The  sides  of  the  triangle 
are  AB  =  7  inches,  AC  =  7  inches, 
BC  =  5  inches.  Find  the  radii  of  the 
circles. 

9.  In  the  figure  AQ=  AP,  BQ=  BR, 
CR  =  CP.  Find  AP,  BQ,  and  CR,  know- 
ing that  AB  =  6  inches,  BO  =  8  inches, 
CA  —  9  inches. 

10.  The  expression  ax*  +  bx2  +  ex  +  5  is 
equal  to  10  when  x=l,  to  15  when  x  =  2, 
and  to  20  when  x  =  3.  Find  the  values  of 
a,  6,  and  c. 

Hint.     Substitute  1,  2,  and  3  successively  for  x.    The  resulting  equa- 
tions are  a  +  b  +  c  +  5  =  10. 
8a  +  4fc-f2c-  +  6  =  16. 
27  a  +0  b  +  8  c  +  5  =  20. 


Problems  with  Three  Unknown  Numbers      277 

11.  Find  the  values  of  I,  m,  and  n  in  the  equation  -  +  -  -f 

l      m 

—=s  1,  if  it  is  satisfied  by  x  =  1,  y  =  2,  2  =  3;  x  =  2,  y  =  —  1, 
n 

z  =  3,  and  <c  =  —  3,  ?/  =  2,  2  =  1. 

12.  The  expression  a#  -f  by  +  C2  =  6  is  satisfied  by  x  —  1, 
y  =  2,  *as3;  a;  =  2,  ?/  =  3,  2  =  1;  a  =  3,  #  =  2,  2  =  1.  Find  the 
values  of  a,  6,  and  c. 

13.  The  sum  of  the  angles  of  a  triangle  is  180°.  Find  the 
number  of  degrees  in  each  of  the  angles  of  a  triangle  if  the 
sum  of  the  first  and  second  angles  and  also  the  sum  of  the  first 
and  third  angles  is  100°. 

14.  Find  the  number  of  degrees  in  each  angle  of  a  triangle, 
if  the  first  angle  is  twice  the  second  and  three  times  the  third 
in  value. 

15.  The  sum  of  the  angles  of  any  quadrilateral  is  360°. 
Find  the  number  of  degrees  in  each  angle  of  a  quadrilateral, 
if  the  sum  of  the  first  and  second  angles  is  160°,  the  sum  of 
the  first  and  third  angles  is  210°,  and  the  sum  of  the  first, 
third,  and  fourth  angles  is  220°. 

16.  Find  the  number  of  degrees  in  each  angle  of  a  quadri- 
lateral, if  the  sum  of  two  opposite  angles  is  200°  and  their  dif- 
ference is  46°,  and  the  difference  of  the  other  two  opposite 
angles  is  30°. 

17.  Three  boys  have  together  88  cents ;  the  first  two  have 
50  cents,  while  the  first  and  third  have  62  cents.  How  much 
has  each  boy  ? 

18.  Hard  phosphor  bronze  for  machine  bearings  contains 
equal  amounts  of  phosphor  tin  and  antimony,  with  9  times  as 
much  copper  as  of  both  other  metals.  How  much  of  each 
metal  is  there  in  120  pounds  of  bronze  ? 


278  Linear  Simultaneous  Equations 

19.  Bronze  medals  are  usually  an  alloy  of  copper,  tin,  and 
zinc.  The  value  of  100  pounds  of  this  material  is  $  18.57,  and 
the  tin  costs  16  times  as  much  as  the  zinc.  How  many 
pounds  of  each  metal  are  there  if  copper  is  18  ^,  tin  30  $,  and 
zinc  5  ^  a  pound  ? 

20.  An  aluminum  alloy  for  crank  cases  in  automobiles  is 
made  of  aluminum,  copper,  and  tin,  and  is  worth  about  $  24.50 
a  hundred  pounds.  The  aluminum  costs  20^  a  pound,  copper 
20^  a  pound,  and  tin  35^  a  pound.  The  value  of  the  tin  in 
100  pounds  is  8J  times  that  of  the  copper.  Find  the  number 
of  pounds  of  each  in  100  pounds. 

21.  German  silver  is  made  of  equal  parts  of  copper,  zinc, 
and  nickel,  and  is  worth  about  25  ^  a  pound.  The  value  of 
the  nickel  is  9  times  that  of  the  zinc  and  2  pounds  of  copper 
and  1  pound  of  zinc  are  worth  as  much  as  1  pound  of  nickel. 
Find  the  value  of  1  pound  of  each. 


XV.    SQUARE  ROOT 

388.  Power.  Square  Root.  A  power  of  a  number  is  the 
product  that  arises  from  using  the  number  one  or  more  times 
as  a  factor.  The  second  power  is  the  square  of  the  number, 
and  the  number  itself  is  the  square  root  of  its  second  power. 

Thus,  9,  ra2,  a2  -+■  2  ab  +  b2  are  the  squares  of  3,  ra,  and  a  +  b,  respec- 
tively, and  3,  ra,  and  a  +  b  are  the  square  roots  of  9,  m\  and  a2  4-  2  ab  + 
b2  respectively. 


Since  the  square  root  of  a  number  is  one  of  the  two 
equal  factors  of  a  perfect  second  power,  numbers  that  are  not 
exact  squares,  strictly  speaking,  have  no  square  roots.  They 
do,  however,  have  approximate  square  roots  and  these  approx- 
imate square  roots  can  be  found  to  any  required  degree  of 
accuracy. 

390.   52  =  25  and  (-  5)2  =  25.     Therefore  25  has  two  square 
roots.     Compare  a2  and  (—  a)2=  a2. 

Thus  it  appears  that  any  number  has  two  square  roots. 

1.    V4  =  2  and  -  2. 

For  convenience,  we  write  Vi  =  ±  2,  and  read  "  the  square  roots  of  four 
are  positive  and  negative  two." 


2.    Va2  =  ±a.  3.    V4a?y=  ±2  xhj. 

ORAL  EXERCISE 

391.   Find  the  square  roots  of  the  following  numbers . 

1.  81.  4.   .25  a4.  7.   tItPV0. 

2.  196.  5!   ayz8.  8.   .01 1/&. 

3.  25  a2.  6.    144  m6.  9.   -^a^14. 

279 


280  Square  Root 

Find  the  square  roots  of  the  following  numbers : 

10.  4  ay.  12.    2.25  x6.  14.    9  ft12?/6- 

11.  .0144  a2.  13.   1.21  a464.  15.   £fa466. 

392.  Square  Roots  of  Monomials  by  Factoring.  The  square 
roots  of  a  perfect  square  can  be  found  by  inspection  if  the 
number  can  readily  be  factored  into  prime  factors. 

1.  Find  the  square  roots  of  2916. 
Solution.    2)2916 

2)1458  

3)  729       .-.  V29l6  =  V22  .  32  .  92  =  ±  2  •  3  ■  9  =  ±  54. 
3)  243 
81 

2.  Find  the  square  roots  of  5184. 
Solution.     5184  =  34  •  26. 

.-.  V518l  =  ±(32.23)=±72. 

3.  Similarly 

V441a264  =  V32.72.a2-64  =  ±3-7.a.&2  =  ±21  a62. 

EXERCISE 

393.  Find  the  square  roots  by  factoring  : 

1.  1225.  7.  784  a262.  13.  43.56  a6. 

2.  2916.  8.  1764  aAb\  14.  .0004  a2. 

3.  2401.  9.  15625  x\  15.  .0121  afy. 

4.  4761.  10.  98.01a6.  16.  a2  +  2  ab  4-  b\ 

5.  7744.  11.  23.04  afyV.  17.  a2-2a  +  L 

6.  5184.  12.  .0841  x\  18.  4(a2-2a  +  l). 

SQUARE   ROOT   OF   POLYNOMIALS 

394.  Since  a2  +  2a6  +  &2=(«  +  by, 

.-.  Va2  +  2  ab  +  b2  =  a  +  b. 

In  order  to  find  the  square  root  of  an  algebraic  expression 
when  it  is  not  so  evident,  as  in  this  case,  a  systematic  method 
must  be  followed. 


Square  Root  of  Polynomials  281 

1.   Find  the  square  root  of  a2  -f  2  ab  +  b2. 

a2  +  2ab  +  b2\  a  +  b 
Subtract  a2  a2 

Trial  divisor,  2  a 


2ab  +  b2 
2ab  +  b2 


Complete  divisor,  2  a  -f-  b 
Multiply  by  b, 
Subtract  2  ab  +  b2 

1.  The  square  root  of  a2  is  a,  the  first  term  of  the  root. 

2.  Subtract  a2,  giving  the  remainder  2  aft  +  ft2. 

3.  Since  2  a&  is  obtained  by  taking  twice  the  product  of  the  first  term 
of  the  binomial  a  +  b  by  the  second  term,  reversing  the  process  and  di- 
viding 2  ab  by  2  a,  or  twice  the  part  of  the  root  already  found,  gives  the 
second  term,  b,  of  the  root.    Hence  2  a  is  used  as  a  trial  divisor. 

4.  Add  b  to  2  a  to  form  the  complete  divisor,  2  a  +  &. 

5.  Multiply  2  a  4-  6  by  b  and  subtract. 

The  above  process  can  easily  be  extended  to  extract  the  square  root  of 
any  polynomial. 

395.   To  extract  the  square  root  of  a  polynomial : 

1.  Arrange  the  terms  in  ascending  or  descending  powers  of  some 
letter. 

2.  Take  the  positive  square  root  of  the  first  term  of  the  polynomial 
as  the  first  term  of  the  root  and  subtract  its  square  from  the  given 
polynomial. 

3.  Take  twice  the  part  of  root  already  found  for  the  first  trial  divisor 
and  divide  the  first  term  of  the  remainder  by  it.  Take  the  quotient  as 
the  second  term  of  the  root. 

4.  Add  the  quotient  just  found  to  the  trial  divisor  to  form  the  com- 
plete divisor.  Multiply  the  complete  divisor  by  the  last  term  of  the 
root  and  subtract  the  product  from  the  first  remainder. 

5.  If  the  second  remainder  is  not  zero,  take  twice  the  part  of  the  root 
already  found  for  a  second  trial  divisor,  and  divide  the  first  term  of  the 
remainder  by  the  first  term  of  the  trial  divisor  to  find  the  next  term  in 
the  root.  Add  this  term  to  the  trial  divisor  to  form  the  second  complete 
divisor  and  proceed  as  before  until  the  remainder  is  zero,  or,  if  the  poly- 
nomial is  not  an  exact  square,  until  the  required  number  of  terms  in  the 
root  has  been  found. 


282 


Square  Root 


Examples 

1.  Extract  the  square  root  of  9x*  -f  6x*y  —  29  x2y2  —  10xyz 
+  25y\ 

The  square  root  of  9a?*  is  3ce».       9a*  +  6a$y  —  29  a^2  —  10a*#3  +  25y*| 3 a*  +  «•?/  -  5 y2 

Subtract  (3 a*)*.  9a^ 

Trial  divisor,  6  a*. 

Complete  divisor,  6 as*  +  »y. 

Multiply  by  xy. 

Subtract  6a%  +  x*y*. 

New  trial  divisor,  6  a'*  -f-  2  a?y. 

New  complete  divisor,  6  a:2  +  2xy  —  5y2 

Multiply  by  -  5  y*. 

Subtract  -  30  a-2^  -  Mxy*  +  25  y*. 

Therefore  the  square  root  of  the  given  expression  is  3  x2  -f  xy  —  5  y2. 
—  (3  a;2  +xy  —  by2)  is  also  a  square  root  of  the  expression.       (Why  ?) 

2.  Extract  the  square  root  of  £x*  —  12  cc5  +  13  a;4  —  14  a3  + 
13a;2_4a;-r-4. 

In  practice  we  usually  abbreviate  the  work  somewhat,  as  in 
the  following : 

4x6  -  12  x5  +  lSx*  -  Uxs  +  13s2  -  4  a?  +  4|2x3  -  3a2  +  x  -  2 

4X6 


6a%  -29cety> 

6  asty  +      a^2 

-80<rty*  -  10  try*  +  25  y* 

A 

-  30 »'2iy2  _  io  xy3  +  25 y* 

4x*-3x* 


-  12  ib6  +  13  a;4 

-  12  x5  4-    9  x* 


4  x3  -  6  x2  +  x|4  s*  -  14  xs  +  13  x2 

|4x4-    6a;3-r       a2 
4  x3  -  6  z2  +  2  x  -  2 


-  8  a:3  +  12  x'2  -  4  ac  +  4 

-  8  xs  +  12  sc2  -  4  x  +  4 


EXERCISE 

I.   1.   How  can  a  square  root  be  checked? 
Find  the  square  roots  of  the  following  expressions: 


2.  m4  —  2  m2n2  +  n4. 

3.  4a!4-12a%3  +  9i/6. 

4.  25a6  +  4  6l4  +  20a367. 

5.  36z2-36az+9z2. 

6.  9  a2x2  -  24  aa  +  16. 


7.  4-12z  +  9z2. 

8.  ia8  +  |&2-|a4&. 

9.  ^m10  +  fm5rc  +  f!»i2. 

16     10^25 


Square  Root  of  Arithmetical  Numbers        283 

E!  +  ^4-j£.  12-   49a;4-}-121?/l-154a;2t?/2. 

9       6      16  i3.   25  a2  4- 60  a&a;  4- 36  &2z2. 

14.  49a6-42a5  +  79a4-30a34-25a2. 

15.  9a2  +24 a3  +  46a4  +  40a5  +  25a6. 

16.  9a2  — 24a3 -14a4 +  40a5  4- 25a6. 

17.  4  -  12a  +  25a2- 24a3  +  16a4. 

18.  16a6 -16a5  +  20a4  +  4a +  1. 

19.  a2-4a&  —  4ao  +  86c  +  462  +  4c2. 

20.  a2 -f  6ax  +  6  ay  +  18a*/  +  9y2  +  9a2. 

21.  4a2-12a^  +  16a,-z-24?/z4-9#24-16z2. 

22.  16a2-40a&  +  24ac-30&c4-2562  +  9c2. 

23.  16a2  -  24a3  4-  25a4  -  20a5  +  10a6  -  4a7  4-a8- 

24.  4  a2  4-  9y2  4-  16z2  4-  25-v2  —  12  xy  4- 16  xz  -  20aw  -  24  yz 
30  yv  —  40  z-v. 

a2     4a6     462     ac     Sbc     9c2 
9       15       25       2  +   5        16  ' 


25. 


SQUARE    ROOT   OF   ARITHMETICAL    NUMBERS 
ORAL  EXERCISE 

397.   Following  are  the  squares  of  some  numbers  : 
12  =  l.  1Q2  =  ioo.  1002  =  10,000.  lOOO2  =    1,000,000. 


32  =  9. 

II2  =  121. 

1012  =  io,201. 

10012=    1,002,001. 

92  =  81. 

992  =  9801. 

9992  =  998,001. 

99992  =  99,980,001. 

By  comparing  these  numbers  and  their  squares,  answer  the 
following  : 

1.  How  many  figures  are  there  in  the  square  of  a  number  of 
one  figure  ?  of  two  figures  ?  of  three  figures  ?  of  n  figures  ? 

2.  How  many  figures  are  there  in  the  square  root  of  121  ? 
of  1521  ?  of  12,100*? 

3.  Use  the  facts  brought  out  in  examples  1  and  2  to  explain 
why  we  point  numbers  off  into  periods  of  two  figures  each 
when  extracting  square  roots. 


284  Square  Root 

4.  How  many  figures  are  there  in  the  square  root  of  a 
number  of  5  figures  ?  of  6  figures  ?  of  7  figures  ?  of  2  n  fig- 
ures ?  of  2  7i  —  1  figures  ?  of  2  n  +  1  figures  ? 

398.  The  whole  process  of  extracting  the  square  root  of  an 
arithmetical  number  is  shown  in  finding  the  square  root  of 
«2  +  2  ab  +  &2  (§  394). 

Since  372  =  (30  +  7)2  =  302  +  2  •  30  •  7  +  72  =  1369,  we  can, 
by  reversing  the  process,  find  the  square  root  of  1369.  Arrang- 
ing as  in  §  394,  we  have 

3024-2.30-7  +  72[30  +  7 
302 


Trial  divisor,  2  .  30 
Complete  divisor,  2  •  30  4-  7 


2  .  30  •  7  +  72 
2  .  30  •  7  +  72 


The  square  root  of  the  first  term  is  30.  After  subtracting  30'2,  the  re- 
mainder is  2  •  30  •  7  +  72. 

The  trial  divisor,  2  •  30,  is  contained  in  the  first  part  of  the  remainder 
7  times.  Then  the  complete  divisor  2  •  30  +  7  is  formed.  The  complete 
divisor  is  contained  exactly  7  times  in  the  remainder. 

.-.  30  +  7,  or  37,  is  the  square  root  of  302  +  2  .  30  •  7  +  T\  or  1369. 
—  37  is  also  a  square  root  of  1369. 

399.    This  is  equivalent  to  the  following: 

37 


13'69 

302  = 

9  00 

2  •  30  =  60 

4  69 

30  +  7  =  67 

4  69 

Trial  divisor, 
Complete  divisor,  2 

First  separate  1369  into  periods  of  two  figures  each.  (Why  ?) 
Since  9  is  the  largest  perfect  square  in  13,  the  square  root  of  1369 
lies  between  30  and  40.  Therefore  3  is  the  first  figure  of  the  root. 
Subtracting  900  and  using  2  •  30  as  a  trial  divisor  (why  ?),  we  find  that 
the  next  figure  of  the  root  is  7.  Completing  the  divisor  by  adding  7 
(why?),  we  find  that  it  is  contained  in  the  remainder  exactly  7 
times. 


Square  Root  of  Arithmetical  Numbers  285 

400.  To  find  the  square  root  of  a  number : 

1.  Separate  the  number  into  periods  of  two  figures  each  beginning  at 
the  decimal  point. 

2.  Write  the  positive  square  root  of  the  largest  perfect  square  in  the 
left-hand  period  as  the  first  figure  of  the  root. 

3.  Subtract  the  square  of  the  first  figure  of  the  root  from  the  left-hand 
period  and  annex  the  second  period  of  the  number. 

4.  Form  a  trial  divisor  by  doubling  the  part  of  the  root  already  found 
and  annexing  one  cipher. 

5.  Divide  the  remainder  by  this  trial  divisor. 

6.  Write  the  quotient  as  the  next  figure  of  the  root  and  add  the 
quotient  to  the  trial  divisor  for  a  complete  divisor. 

7.  Multiply  the  complete  divisor  by  the  last  figure  obtained  in  the 
root  and  subtract  the  result  from  the  last  remainder. 

8.  If  there  are  more  than  two  periods  in  the  number,  annex  the  next 
period  to  the  remainder  and  repeat  steps  4  to  7  until  there  is  no  remain- 
der, or  in  case  the  number  is  not  a  perfect  square,  until  the  required  num- 
ber of  decimal  places  is  obtained. 

Examples 

401.  1.    Find  the  square  root  of  4719.69. 


47'19.'69 

36 

128 


136.7 


.7 

The  first  trial  divisor  is  120,  and  the  complete 

divisor  is  128. 

11    XV. 

,ft  24  The  second  trial  divisor  is  136.0,  and  the  com- 

o^aq  plete  divisor  is  136.7. 

Therefore  V4719.69  =  ±  68.7. 


95.69 


2.   Find  the  square  root  of  41209. 

4'12'091203  Here  tne  ^rat  tria*  divisor,  4(*>  is  larger  tDan  the 

remainder  12.     Put  a  zero  in  the  root  and  bring 

403 1    12  09  down  another  period.     The  trial  divisor  now  be- 

1   12    '  comes  400,  and  the  next  figure  in  the  root  is  3.    The 

square  roots  are  ±  203. 


286 


Square  Root 


3.   Find  the  square  root  of  2  to  three  decimal  places. 
2.'00'00'00|  1.414+ 

After  pointing  off  into  periods,  the  decimal 
point  may  be  neglected.  How  will  the  num- 
ber of  figures  to  the  left  of  the  decimal  point 
in  the  answer  compare  with  the  number  of 
periods  to  the  left  of  the  decimal  point  in  the 
number  ? 

604 

EXERCISE 

402.    1.   In  extracting  the  square  root  of  a  number,  why  do 
we  separate  the  number  into  periods  of  two  figures  each  ? 

2.  Will  the  division  of  the  remainder  by  the  trial  divisor 
always  give  the  next  figure  of  the  root  ?     Explain  your  answer. 

3.  Square  the  result  in  example  3,   §  401,  and  add  the 
remainder  ;  that  is,  1.4142  +  .000604. 


24 

100 
96 

281 

400 
281 

2824 

11900 
11296 

Extract  the  square  roots  of  the  following 


4.  4096. 

5.  6241. 

6.  161.29. 

7.  2.3716. 

8.  61504. 

9.  1108.89. 

10.  277729. 

11.  13456. 

12.  30276. 


13.  119025. 

14.  .093025. 

15.  .007569. 

16.  .098596. 

17.  12.8881. 

18.  11669056. 

19.  6504.4225. 

20.  .83064996. 

21.  95121009. 


22.  101062809. 

23.  .00917764. 

24.  1400.2564. 

25.  .00762129. 

26.  .0009979281. 

27.  100020001. 

28.  29495761. 

29.  64128064. 

30.  44105040144. 


Find  the  square  roots  of  the  following,  to  two  decimal  places  : 

31.  2.2.  35.    7.  39.   3.666. 

32.  3.  36.    8.  40.    27.1917. 

33.  5.  37.   3.1416.  41.   391. 

34.  6.  38.    210.  42.    10.004. 


Square  Root  of  Arithmetical  Numbers         287 


43.    40.003. 

45.    V_. 

47.    80. 

44.   J/- =  6.5. 

46.    5|. 

48.    82. 

By  reducing  to  a  decimal  find  the  square  roots  to  three  deci- 
mal places  in  examples  49  to  53 : 

49.    f.  50.    2f.  51.    5f  52.    16|.  53.    f 

By  first  making  the  denominator  a  perfect  square  find  the 
square  roots  in  examples  54  to  61  to  three  decimal  places : 

54.    f 

/5 J30=  V30  =  5.477 

'36        6  6 

to  three  decimal  places. 


Suggestion.     -v-=A/—  =  —  : — =  -913.     This  result  is  correct 


55.    TV  56.    f  57.    f.  58.11  59.    #. 

60.    V4  +  V2  61.       fe+VO 

2       *  X        2 

62.  To  find  the  approximate  square  root  of  \,  why  is  it 
better  to  use  \  than  \  ? 

403.  It  is  sometimes  desired  to  find  the  square  root  of  an 
algebraic  expression  that  is  not  a  perfect  square  correct  to  a 
certain  number  of  terms.  The  process  does  not  differ  from 
that  used  when  the  expression  is  a  perfect  square. 

Find  the  square  root  of  1  -f-  x,  in  the  ascending  powers  of  x 
to  three  terms. 


1  +  x 

1 

2  +  ix 


l  +  ±x-±x2 


x 

x  +  \x* 


I-2 


1/^3   1      r/A 


To  check,  square  the  result  and  add  the  last  remainder. 


288  Square  Root 

EXERCISE 

404.   Find  the  square  roots  of  : 

1.  1  —  a  to  three  terms  and  check. 

2.  1  —  a2  to  three  terms  and  check. 

3.  4  +  x2  to  four  terms. 

4.  1  -f  x  +  x2  to  three  terms  and  check. 

5.  a2  +  x  to  three  terms. 

6.  1  -f  4  a2  to  three  terms. 

7.  1  +  x  —  x2  to  three  terms. 

8.  13  x2  —  3  x3  +  4  x4  —  12  x  +  4  to  three  terms  and  check. 

9.  a6  +  4a56-2a4&2-12a363  +  9a2&4. 

10.  16  a6  -  24  a5?/ +  65  a4?/2 -42  a3?/3 -f  49  a2?/4. 

11.  a2  +  4a&  —  2ac+4&2  —  46c-f  c2. 

12.  4  a4  -  12  a362  -  4  a263  +  9  a264  +  6  abb  +  66. 

13.  a4-2a3  +  3a2-2a  +  l. 

14.  1  4-  a2  —  2  a4  +  a6  4-  2  x  —  2  a8. 

15.  2  mw  +  £>2  -j-  2  np  +  a'2  +  2  mp  +  m2. 

16.  l  +  4?/2  +  a2  -4?/ +  2  a  —  4  ay. 

17.  4  +  13  a2  +  9  a4  -4  a  -6  a3. 

18.  18a2+a4  +  l-8a3-8a. 

19.  a262+2a26  +  a2-2a&2-2a&  +  &2. 
m4      ?i6      m2/)4      m2nz      p*      n3p4 

20-  t+9+^ — r+«r  6 

01  a2     4az ■    4z2  ,  6ga  ,  9g2     12^z 
y2      uy        u2        vy        vl         uv 


Square  Root  of  Arithmetical  Numbers         289 

Find  the  square  root  oj  each  of  the  following  polynomials  to 
three  terms  : 

23.  1  +  x2.  26.    1  +  x  +  x2  +  a3  +  x4. 

24.  1  —  4  a2.  27.    1  —  a  +  a2  —  Xs  +  xA. 

25.  a^  +  x2.  28.    1  +  a?  +  x4  +  a6  +  «*• 

i^md  £/ie  fourth  root  of: 

29.  -^  to  two  decimal  places. 

Suggestion.     Take  the  square  root  of  the  square  root 

30.  aj*  +  4B3  +  6<e2  +  4aj+l. 

31.  14641.  32.   3  to  two  decimal  places. 

33.  In  the  figure  x2  =  a2  +  b2.      Find  the       b_ 

value  of  x  to  two  decimal  places  when  a  =  3 
inches  and  6  =  7  inches. 

34.  As  in  example  33  find  the  value  of  x  when  a  =  b  =  10. 

35.  Find  t  correct  to  one  decimal  place,  if  £=-v/ — ,  when 
d  =  100  and  g  =  32.  ^ 

36.  Find  s  in  *  =  Vl  -  c2,  if  c  =  $ V3. 

37.  Find  s  in  example  36  to  two  decimal  places  if  c  =  \. 


38.  Find  Tto  two  decimal  places  in  T=  Vs(s— a)(s—  b){s— c), 
where  a  =  6,  6  =  7,  c  =  9,  and  s  =  i(a  +  6  +  c). 

39.  Find  T  in  example  38  if  a  =  b  =  c  =  8. 

40.  Would  the  value  of  V5  to  three  decimal  places  give  the 

value  of  10  V5  correct  to  three  decimal  places  ?    Would  it  give 

V5 

the  value  of  correct  to  three  decimal  places  ? 

10  F 


41.   Find  y  if  y  =  20  vl  +  V2  +  V3.     Get  the  answer  correct 
to  two  decimal  places. 


XVI.  QUADRATIC  EQUATIONS 

405.  Quadratic  Equations.  A  quadratic  equation,  or  an  equa- 
tion of  the  second  degree  containing  one  unknown  number,  is 
an  equation  that,  when  reduced  to  its  simplest  integral  form, 
contains  the  second  power  of  the  unknown  number,  and  no 
higher  power  than  the  second. 

Thus,  2  x2  +  3  x  =  7,  x2  —  5  =  0,  3  y2  —  5  y  =  0  are  all  quadratic  equa- 
tions, but  (x  —  l)(x  +  2)  =  x2  -f  7  is  not  a  quadratic  equation,  for,  when 
reduced  to  its  simplest  form,  it  becomes  x  —  9  =  0,  a  linear  equation. 

406.  Absolute  Term.  The  term,  or  group  of  terms,  not  con- 
taining the  unknown  number  is  the  absolute  term  of  the  equa- 
tion. 

Thus,  in  the  equation  3x2  +  5x  —  7  =  0,  —  7  is  the  absolute  term. 

407.  Incomplete  Quadratic.  If  a  quadratic  equation  does  not 
contain  a  term  of  the  first  degree  in  the  unknown  number,  as, 
x2  —  5  ss  0,  or  if  the  absolute  term  is  0,  as  3  x2  —  5  x  =  0,  it  is 
an  incomplete  quadratic  equation. 

408.  Complete  Quadratic.  If  a  quadratic  equation  contains  a 
term  of  the  second  degree  in  the  unknown  number,  a  term  of 
the  first  degree  in  the  unknown  number,  and  an  absolute  term, 
the  equation  is  a  complete  quadratic  equation. 

Thus,  2  x2  +  3  x  =  7  is  a  complete  quadratic  equation. 


Incomplete  Quadratic  Equations  291 

INCOMPLETE  QUADRATIC  EQUATIONS 

409.  Solution  of  the  Quadratic  Equation  Lacking  the  Term  Con- 
taining the  Unknown  of  the  First  Degree. 

1.    Solve  x2  -  25  =  0.  2.    Solve  x*  -  7  =  0. 

Solution,     x2  -  25  =  0.  Solution,    x2  -  7  =  0. 

x2  =  25.  *  =  7- 

x  =  ±5.  ac=V7  =±2.65+. 

410.  To  solve  a  quadratic  equation  in  which  the  first  degree  term  in 
x  is  lacking : 

1.  Clear  of  fractions,  expand,  transpose,  and  reduce  the  equation  to 
the  form  x2  =  k. 

2.  Extract  the  square  root  of  both  members,  using  the  double  sign  be- 
fore the  root  of  the  second  member. 

3.  If  the  second  member  is  not  a  perfect  square,  find  its  approximate 
value  to  any  required  number  of  decimal  places. 

Examples 

1.  Solve  (3  x  +  1.5)(3  x  -  1.5)  =  54. 
Solution. 

(3z  +  1.5)(3z-1.5)=54. 

9  x2  -  2.25  =  54.      (Expanding. ) 

9x2  =  56.25.     (Transposing  and  collecting.) 

2_  56.25 
x — . 

7  5 
X  as ± "-J-  =  ±  2.5.     (Extracting  the  square  roots 

of  both  members. ) 

Check.     Substitute  the  answers  in  the  original  equation.     Thus, 

(3  x  2.5  +  1.5)(3  x  2.5  -  1.5)  =  9  x  6  =  64. 

Let  the  student  verify  the  negative  answer. 

2.  Find  correct  to  two  decimal  places  the  roots  of 

x  -  10  _     7 
6         3  +  10' 
Solution.  s2  -  100  =  42.     (Clearing  of  fractions.) 

x2  =  142. 

x=  ±  11.916.    (Extracting  square  root.) 
The  closest  approximation  to  two  decimal  places  is  x  =  ±  11.92. 


292  Quadratic  Equations 

411.  The  verification  of  approximate  answers  may  become 
tedious.  Approximate  verifications  will  generally  serve  to 
detect  large  errors  in  answers.  In  the  above  example,  12 
is  a  close  approximation  to  the  answer.  Putting  12  for  x  in 
both  members  we  should  have  -|  in  the  first  member  and  -fa  in 
the  second.  These  values  are  not  greatly  different,  and  the 
answer  is  probably  correct.  The  most  satisfactory  verification 
in  this  case  is  to  go  carefully  over  the  work  again. 


EXERCISE 

412.   Solve  the  following  equations: 

1.   a? -169  =  0.  7.  x*  +  4  =  13. 

^_2  =  40  8-  «2  =  30276. 

40  9-  a2-f=8£. 

3.    13a*-19  =  7^  +  5.  io.  3a*  =  7 -a* 

3  **  11.  -=27. 

5.  x*  =  6£.  3 

6.  z2-a2  =  2a  +  l.  12.  3 a2  =  210.25. 

13.  (2x  +  7)(5x- 9)+(2a  -  7)(5a  +  9)=  1874. 

,.     2m  — 1       ra  — 5  -,/»  4  4  1 

14.    —  = -.  16. 


15. 


ra-2       3m  — 2  x  +  3     a  -  3         3' 

25  +  3^13  +  x  1?    3a2     15v2  +  8  =  oa?     3 

9  + a      47  —  a*  '4  6 


413.   Solution  of  the  Incomplete  Quadratic  with  Absolute  Term 
Lacking. 

Solve  3  a*  _  5  x  =  0.     (See  §  239.) 

Solution.  (3x  —  6)a;  =  0.     (Factoring  the  left  member.) 

Sx-5  =0otx  =  0.     (§238.) 
x  =  £  or  0. 


Incomplete  Quadratic  Equations  293 

414.    To  solve  a  quadratic  equation  in  which  the  absolute  term  is 
lacking : 

1.  Clear  of  fractions,  expand,  transpose,  and  simplify  until  the  equa- 
tion is  in  the  form  ax2  +  bx  =  0. 

2.  Solve  the  equation  by  factoring. 


1.   Solve  x  +  2  = 


Examples 

7z-4 


Solution.  x2  —  4  =  7  x  —  4.     (Clearing  of  fractions.) 

x2  —  7  *  =  0.     (Simplifying.) 
x(x-7)=0. 
x  =  0orx-7=0.     (Why?) 
x  =  0  or  x  =  7. 
Check  the  answers  mentally. 

2.   Solve  (3  a;  -  5)2  -  (2  sc  -  3)2  =  16. 

Solution. 

9  x2  -  30  x  +  26  -  4  x2  +  12  x  -  9  =  16.      (Expanding.) 

5  x2  —  18  x  —  0.     (Transposing  and  collecting. ) 
x(5x  -  18)=  0.     (Factoring.) 
x  =  0  or  5  x  -  18  =  0.     (Why  ?) 
x  =  0  or  Jj£. 

EXERCISE 
415.   Solve: 

1.  x2-9x  =  0.  4.  aaj2+&c  =  0. 

2.  3z2  +  4z  =  0.  5.  (3z-7)2-(5a-3)2=40. 

3.  5m2-3m  =  0.  6.  (a?  +  5)2  +  (ic-3)2  =  34. 

7.  2(x  +  3)2  -  (a?  -  3)2  =  9. 

8.  (a-a)2+2(<e  +  a)  =  a2  +  2a. 

9.    (z-5)2-(2a;-3)2=16.        ^     2£!  +  2  +  6p==2 
10.   (m-l)(m  +  l)=2m-l.  '5  7 

3       2  J^  t/-10 


294  Quadratic  Equations 


Solve : 

14.    -^-  =  1 Kr'  16. 

2+1  Z  — 1 


2a 

-1 

i 

2z  +  3 

1 

2a 

X- 

+  1 
-1 

i 

a +  2 
-3 

2 
1 

X  - 

-2 

X 

-4 

4 

15.    ^+_3  +  3s-2  =  1  ^ 

a?  +  5       x  —  5 


COMPLETE  QUADRATIC  EQUATIONS 

416.  The  complete  quadratic  equation  has  been  solved  by- 
factoring.  The  student  should  carefully  review  §§  237  to 
239. 


EXERCISE 

417.  Solve  the  following  by  factoring : 

1.  x*-3x  +  2  =  0.  5     «,10  =  11> 

2.  x*  +  5x  +  6  =  0.  2     6a?      6 

6.   3«2-5a;=10+2a;2-2ic. 

3.  2*2  +  *_3  =  0.  7    (^  +  5)2=2(^  +  3)2-17. 

4.  m2- 2m -24  =  0.  8.   aj(aj  -  1)  =  380. 
9.    (2jp  -  8)2  =  4(3  p  +  25)  +  12. 

10     z  +  2 _      36  j^  14.  4(?-2-l)+r  +  l  =  0. 

2  +  3      (z  +  3)2  15.  5(a2-4)-(a-2)=0. 

11.   Kv  +  5)=3(-2/-5).  16.  S2_8  =  7s-14. 

&  =  x_x.1  17-  3w2  +  w  =  10. 

'632  18.  (5x-2)(6x*-x-2)=0. 

13.    (2m)2-5(2m)-6  =  0.  19.  x^-5x  +  A  =  0. 
20.   What  is  a  root  of  an  equation  ? 

418.  Completing  the  Square.  In  §  204  we  learned  that  any 
one  of  the  three  terms  of  a  perfect  trinomial  square  can  be  sup- 
plied if  we  know  the  other  two  terms. 


Complete  Quadratic  Equations  295 

ORAL  EXERCISE 

419.  In  each  of  the  following  supply  the  proper  number  in  the 
parenthesis  to  make  a  perfect  trinomial  square,  and  find  the  square 
root  of  the  trinomial : 

1.  x2  +  2ax  +  (    ).  8.   x2+6x  +  (    ). 

2.  4#2  +  4a+(    ).  9.   x2_$x+(    y 

3.  4a4  +  4a;2+(     ).  10.  x2  +  x  +  (     ). 

4.  4a4+(     )+4#2.  11.  a2-3ar+(     ). 

5.  9  +  6z+(     ).  12.  a2-?+(    ). 

6.  (     )  +  6*  +  l.  2      * 

7.  a2  +  2z  +  (     ).  13'  x2~a+(    }* 

14.  State  a  rule  for  completing  the  square  in  expressions  of 
the  form  x2  -f-  px. 

420.  The  >form.  Every  quadratic  equation  with  one  un- 
known number  can  be  written  in  the  form  ax2  +  bx  -f  c  =  0. 
This  can  be  further  simplified  by  dividing  through  by  a,  giving 

be  b  c 

x2  -f-  -x  -f-  —  =  0.     By  putting  -  =  p  and =  q,  the  equation 

a        a  a  a 

assumes  the  form  x2  -f  px  =  g.  For  convenience  we  shall  call 
this  the  p-form.  It  requires  that  the  coefficient  of  x2  be  -f-  1, 
and  that  the  absolute  term  be  in  the  second  member  of  the 
equation,  p  and  q  may  be  any  positive  or  negative  numbers, 
integers,  fractions,  monomials,  or  polynomials. 

Examples 
Examples  1  to  5  below  are  in  the  p-fovm. 

1.  x2-7x  =  10-,  p  =  -7,  and  q  =  10. 

2.  aj2  +  |  =  -9;p==i,andg  =  -9. 

3.  x2  =  90 ;  p  =  0,  and  q  =  90. 

4.  x2  +  (a  -f-  6)a;  =  0  ;  p  =  a  +  6  and  g  =  0. 

5.  x2  —  -  =  (&  -  c)2 ;  p  =  -  -,  and  q  =  b2  -  2  be  +  c2. 

ft  ft 


296  Quadratic  Equations 


EXERCISE 

421.  Change  the  following  into  the  p-form,  and  determine  the 
value  ofp  and  q  for  each : 

1.  x2  +  bx  -f-  c  =  0. 

2.  2z2-b  15.9  =  13.6  a;. 

3.  (a?  — 7)(aj  — 5)=0. 

4.  (x-iy  =  a(x*-l). 

5.  c(a-xy+(x-by  =  a2  +  b2. 

6.  ?L^_^+1  =  0. 
4-x       4 

422.  The  Solution  of  the  Complete  Quadratic  Equation  by  Com- 
pleting the  Square.  The  solution  of  a  quadratic  equation  by 
factoring  fails  when  the  factors  cannot  be  found.  The  method 
about  to  be  given  will  solve  in  all  cases. 

1.   Solve  the  equation  x2  -f-  9x  =  10. 

Solution,    x2  +  9  x  =  10. 

x2  +  9 x  +  V  =  if1.     (Adding  *£  to  both  members.) 
x  +  |  =  ±  V-     (Extracting  square  roots.) 
z  =  1  or  -  10. 

Check.         12  +  9.1  =  10. 

(_  10)2  +  9(_  10)  =  100  -  90  =  10. 

How  do  you  determine  that  y  is  to  be  added  ?  Why  do  you  add  it  to 
both  members  ?  Why  do  you  use  the  double  sign  in  the  second  member  ? 
Why  do  you  not  use  the  double  sign  in  both  members  ? 

The  solution  of  example  1  illustrates  the  method  of  solving 
complete  quadratic  equations  by  "  completing  the  square." 
This  equation  was  in  the  j>form  at  first.  The  steps  required 
to  reduce  any  quadratic  equation  to  the  p-form  are  already- 
familiar  to  the  student. 


Complete  Quadratic  Equations  297 

2.    Solve,  getting  the  answers  correct  to  two  decimal  places, 
X2_f_(;r  +  2)2  =  180. 

Solution.  x2  +  x2  +  4x  +  4  =  180.     (Why?) 

2  x2  +  4  x  =  176.     (Why  ?) 
x2  +  2x  =  88.     (Why?) 
x2  +  2x  +  l  =  89.     (Why?) 

x  +  i  =  ±  9.43+.     (Why  ?) 
x  =  8.43+  or  -  10.43+. 

These  roots  can  be*  obtained  to  any  required  degree  of  accuracy  by 
finding  the  square  root  of  89  correct  to  more  decimal  figures. 

423«   To  solve  a  complete  quadratic  equation  : 

1.  Reduce  the  equation  to  the  p-iorm. 

2.  Complete  the  square  of  the  first  member  by  adding  to  both  mem- 
bers the  square  of  one  half  the  coefficient  of  x. 

3.  Extract  the  square  root  of  each  member  of  the  equation  and  solve 
the  resulting  linear  equations. 

Examples 

1.  Solve  6  x2  =  x  +  15. 

Solution.  6  x2  -  x  =  15.     (Why?) 

x*-\x  =  \.     (Why?) 

*-A=±H-     (Why?) 

Check.  6  •  (f  )2  =  \  +  15  or  ^°  =  *£. 

Let  the  student  check  the  other  root. 

2.  Solve  x2  +  ax  =  ac  +  ex. 

Solution.  x2  +  ax  —  ex  =  ac. 

x2  +  (a  —  c)x  =  ac. 

*+(._,),  +  («-«)' =  «,  +  («-«)■  or  «*  +  2f +ig. 
4  4  4 

2  2 

_ _     a  - c  ,  a  +  c 
X  —  -^±     2     ' 
x  =  c  or  —  a. 
Let  the  student  check  mentally. 


298  Quadratic  Equations 

EXERCISE 

424.    Solve  the  following  equations  by  completing  the  square, 
finding  all  roots  correct  to  two  decimal  places: 

1.  a?2  +  2a;=3.  24.  (2a;-15)(3a;+8)=-154. 

2.  x2  -  10  x  =  200.  25.  8a;2  +  2a;-15  =  0. 

3.  tz  +  t  =  12.  26.  20  a;2  +  2a;- 7  =  0. 

4.  x2  -x  =  12.  27.  6(a;2  +  l)=13a;. 

5.  a?  +  3a;  =  10.  28.  3s2-16  =  7s. 

6.  w2  +  3w  =  108.  29.  a?  +  ia;  =  2. 

7.  a2 +  17  a;  =  30.  30.  x2  +  6.51  =  5.2  a. 

8.  a;2-8a;  +  15  =  0.  31.  ?/2  +  .2 y  -  .15  =  0. 

9.  a2-40a  +  111  =  0.  32.  x2  +  6a?  -  2  b2  =  0. 

33.  (a?  —  7)  (a?  —  5)  =40. 

35.  a;2  +  22(a;  +  5)=0. 

36.  (4a;-l)(a;  +  l)=75. 

37.  p(p  —  6)=7p-42. 

38.  4ari  +  (a;-l)2-3a;=31. 

39.  q(q-2)=67. 

An    „      ,1      4a;  +  7 

40.  7s  +  T=       T    ■ 
4        16  a; 

41.  a?(aj  +  l)=-^. 

„0     3  a,/       4\     19 
42'    T^-3)  =  24- 

43.   a;2 -8  a; -14  =  0. 

44.  (x  +  4)  (a?  +  5)  =  2(s  +  2)(a>  +  4). 

45.  (3-2  a>)(l  -  3 a;) (2  -  aj)=*  x(l  —  6  a>)(»  -  2). 

46.  (a;  +  6)  (a?  -  4)  +  (x  +  2)(a?  -  2)  =  56. 


10. 

x2-2Ax  +  .$  =  0. 

11. 

x2-2ax  =  3  a2. 

12. 

c2  +  2c  =  -l. 

13. 

a;2-36a;  +  }&2  =  0. 

14. 

a;2 -32  a;  =  32. 

15. 

2v2  +  3v  =  108. 

16. 

3a;2 -5a;  =  2. 

17. 

6  x2  +  1  =  5  a;. 

18. 

5  ra  —  m2  =  —  50. 

19. 

15  a;2 +  8a;  =  3.75. 

20. 

9a;2 +  17  a;  =  310. 

21. 

Ja^  +  ^Of-A-O. 

22. 

&(7-5)=6. 

23. 

far5 -11a; -15  =  0. 

Complete  Quadratic  Equations  299 

47.  (a-l)2+(a  +  l)  +  (2a  +  3)2  =  29. 

48.  4c2-3c=31-(c-l)2. 

49.  (4-d)(5d  +  l)-d(4-d)=0. 

50.  (7  -  a)a  =(l  -  -V  a?  +  8). 

51.  x(x  +  l)(a>  +  3)-(a>  +  £)(a  +  -J) (a?  +  ±)=  0. 

52.  3a(a  +  l)-(7  +  2a)=0. 

53.  118z-2i22  =  20. 

54.  a2  4-  a&  =  a(a  +  6). 

55.  (a-3  6)(a  +  2&)=6&2. 

56.  (2y-«)(2e-y)  +  (5y  +  2e)e  =  0. 

57.  r(r  +  1.25)  =  .75(r  +  1.25). 

58.  (r  +  3)2+(r  +  5)2  =  514. 

59.  £2  =  l-£. 

60-    A^  +  tJ^^-A—0- 

61.   a;(7a;-l)+^-20^  +  3)+8  =  0. 

e2    4a?~7     2a  +  3_23   2     94 
62'    ~X~+~~9~"-45a;~45' 

63.    M^*l 2 3==()> 

2a-l         2a- 1 

64.  J^-6-|^  =  0.  69.  *:2(*-3Wa?-3:»-l 

a  +  2  3  a; 

65.  ^+lY-?a;  =  3.  70.    10:Z=Z:10-^. 

66.  ?0a  +  l  =  i9.  71.    __3 L_  =  l. 

3        a?      3  2(a?-l)      4  (a?  + 1)      8 

a? +  11     2a?  +  l  =  0  72       4a        g  +  3'=1 

a  +  3        a  +  5  a  —  1         a; 


67. 


68.    -i§ l^_  +  5  =  0.  73.     a2-a  +  3  =  a  +  5 

2  +  3     2  +  10  aj*-4a>  +  5     a-1 


300  Quadratic  Equations 

Solve  the  following  equations  by  completing  the  square,  finding 
all  roots  correct  to  two  decimal  places : 

74    M-2_2(y-2)_2=a   75     J_+^ 1_  =  0. 

y  y+1  x  +  1      x  +  2     x  +  3 

76  x-5      x  +  8         80  ,  =1 
'    s  +  3  +  x_r9-^     2 

77  2*  +  7      3s-2=5 
2  a;  —  3       05  +  1 

425.  1.  In  solving  any  problem  by  means  of  an  algebraic 
equation,  the  student  should  first  carefully  read  the  problem  so 
that  he  can  correctly  translate  the  verbal  language  into  the 
algebraic  language  of  the  equation. 

2.  He  should  then  solve  the  equation  in  the  most  direct  way 
possible. 

3.  He  should  check  and  interpret  the  results  of  the  solution. 
It  should  be  noted  that  the  conditions  of  the  problem,  with 
all  their  restrictions,  cannot  always  be  translated  into  an 
algebraic  equation,  so  that  the  solution  of  the  equation  may 
give  roots  that  do  not  satisfy  the  conditions  of  the  problem. 
See  example  4  following. 

PROBLEMS 

426.  1.  The  area  of  a  circle  is  ttjR2  where  ir  ==  3.1416  and  R 
is  the  radius.  Find  the  radius  of  the  circle  whose  area  is 
78.54  square  inches. 

2.  The  area  of  a  circle  is  100  square  inches.  Find  the 
radius  correct  to  two  decimal  places. 

3.  Find  two  consecutive  integers  if  the  sum  of  their  squares 
is  25. 

Solution.  Let  x  =  the  smaller  number. 

Hence  *  4-  1  =  the  larger  number.     (Why  ?) 
Then  «2  +  (as  +  l)2  =  25,     (By  the  conditions.) 
or  2x2  +  2x  +  1  =  25.     (Why  ?) 
.•.x2  +  z  =  12.     (Why?) 


Complete  Quadratic  Equations  301 

&  +  x  +  i  =  -4?9--     (Why?) 

x  =  3  or  —  4  =  the  smaller  number, 
and  z  +  l  =  4  or  —  3  =  the  larger  number. 
The  answers  are  3  and  4,  or  —  4  and    —  3,  either  pair  of  numbers 
satisfying  the  conditions. 

4.  The  square  upon  the  longest  side  of  a  right-angled 
triangle  is  equal  to  the  sum  of  the  squares  upon  the  other  two 
sides.  In  a  certain  right-angled  triangle  one  of  the  sides  about 
the  right  angle  is  1  inch  longer  than  the  other  and  the  hypote- 
nuse is  5  inches  long.     Find  the  two  sides  about  the  right  angle. 

Solution.  Let  x  =  number  of  inches  in  one  of  the  sides. 

Hence  x  +  1  =  number  of  inches  in  other. 
Then  z2  +  (x  -f  l)2  =  25.     ( Why  ?) 

The  solution  from  this  point  on  is  exactly  the  same  as  in  problem  3, 
but  the  negative  answers  that  were  satisfactory  in  problem  3  have  to  be 
rejected.     The  sides  of  the  triangle  are  3  inches  and  4  inches. 

Note.  In  the  solution  of  applied  problems,  careful  attention  must  be 
given  to  the  interpretation  of  the  answers  obtained.  Sometimes  one, 
sometimes  both  answers  satisfy  the  conditions  of  the  problem.  It  may 
happen  that  neither  of  the  answers  will  satisfy  the  conditions.     (Why  ?) 

5.  By  solving  as  in  problem  3,  find  out  if  there  are  two 
consecutive  integers  the  sum  of  whose  squares  is  32. 

6.  Separate  360  into  two  factors  whose  difference  is  9. 

(This  problem  can  be  solved  by  any  one  of  the  three    equations 

(a)  x(x  -  9)  =  360 ;  (6)  x{x  +  9)  =360;    (c)  x  -  —  =  9.     Explain  and 
solve  each  equation.)  x 

7.  The  sum  of  a  number  and  its  reciprocal  is  -f .  What  is 
the  number  ?  Do  both  roots  of  the  equation  satisfy  the  con- 
ditions ? 

8.  The  area  of  a  rectangle  is  720  square  inches.  The  dif- 
ference of  its  two  unequal  sides  is  12  inches.  Find  the 
dimensions. 

9.  How  long  is  each  side  of  a  square  if  the  diagonal  is 
10  inches  long?     (See  problem  4.) 


302 


Quadratic  Equations 


10.  The  two  unequal  sides  of  a  rectangle  are  in  the  ratio  of  5 
to  12,  and  the  diagonal  is  6.5  inches  long.     Find  the  dimensions. 

Suggestion.  Let  the  number  of  inches  in  the  sides  be  5  x  and  12  x  . 
and  see  problem  4. 

11.  The  area  of  a  rectangle  is  2400  square  inches.  The 
ratio  of  its  two  unequal  sides  is  5  to  12.     Find  its  dimensions. 

12.  The  sum  of  the  areas  of  two  squares  is  233  square 
inches ;  the  sum  of  their  sides  is  21  inches.  Find  the  side  of 
each  square. 

13.  In  the  accompanying  figure  the  shaded  area  is  equal  to 
21.46  square  inches.     Find  the  radius  of  the  circle.     (The  side 

of  the  square  equals  twice  the  radius  of 
the  circle,  and  the  difference  in  their  areas 
is  the  shaded  part.  See  also  the  first  prob- 
lem of  this  set.) 

14.  Find  two  numbers,  one  of  which  is 
double  the  other,  such  that  the  sum  of  then- 
squares  exceeds  the  sum  of  the  numbers 
by  68. 

15.  Find  two  numbers,  one  of  which  is  double  the  other,  if 
the  square  of  their  sum  exceeds  the  sum  of  their  squares 
by  100. 

16.  Find  two  consecutive  numbers  if  the  sum  of  their 
squares  exceeds  the  product  of  the  numbers  by  43. 

17.  If  18  is  divided  by  a  certain  number,  the  quotient  is 
greater  by  1^  than  if  the  divisor  were  increased  by  2.  Find 
the  first  divisor. 

18.  Find  two  consecutive  even  numbers  the  sum  of  whose 
reciprocals  is  ^. 

19.  A  train  makes  a  run  of  280  miles  in  1  hour  and  45  min- 
utes less  time  than  another  train  whose  rate  is  8  miles  an  hour 
less.     Find  the  rate  of  each  train. 

Suggestion.    Remember  that  distance  -4-  rate  =  time. 


Complete  Quadratic  Equations  303 

20.  A  woman  buys  cloth  for  $  8.  Had  she  paid  40  ^  more 
per  yard  she  would  have  received  one  yard  less  for  the  same 
amount.     How  much  per  yard  did  the  cloth  cost  ? 

21.  A  man  bought  a  flock  of  sheep  for  $  75.  If  he  had  paid 
the  same  sum  for  a  flock  containing  3  more  sheep,  they  would 
have  cost  $  1.25  less  per  head.  How  many  did  he  buy,  and  at 
what  price  per  head  ? 

22.  S  =  \gt2  -f  v0t  is  a  formula  much  used  in  physics.  Find 
t  when  S  =  520,  g  =  32,  and  v0  =  24. 

23.  Find  the  value  of  t  in  S  =  ±gt2  +  v0t  when  £  =  100, 
g  =  32,  and  v0  =  0. 

24.  m  :  n  =  x2 :  (a  —  x)2  is  a  relation  used  in  the  study  of 
light.     Find  the  value  of  x  when  m  =  4,  n  =  3,  and  a  =  150  cm. 

25.  A  rope  100  feet  long  is  stretched  around  four  posts  set  at 
the  corners  of  a  rectangle  whose  area  is  576  square  feet.  Find 
the  dimensions  of  the  rectangle. 

26.  The  sum  of  the  two  unequal  sides  of  a  rectangle  is  20 
feet  and  the  diagonal  is  16  feet  long.  Find  the  lengths  of  the 
sides  correct  to  2  decimal  places. 

27.  A  farmer  bought  some  sheep  for  $134.40.  If  each 
sheep  had  cost  him  80^  less,  he  could  have  bought  3  more  for 
the  same  amount.     How  many  sheep  did  he  buy  ? 

28.  A  traveler  made  a  journey  of  630  miles.  He  would 
have  required  4  days  less  to  make  the  journey  had  he  gone 
10  miles  farther  each  day.  How  many  days  did  the  journey 
require,  and  how  many  miles  did  he  travel  each  day  ? 

29.  A  traveler  made  a  journey  of  630  miles.  He  would 
have  required  4  days  more  to  make  the  journey  had  he  traveled 
10  miles  less  each  day.  How  many  days  did  the  journey 
require,  and  how  far  did  he  travel  each  day  ? 

30.  Solve  V=  i  h(S2  +  s2  +  Ss)  for  S  where  V  =  252,  h  =12, 
and  s  =  3. 


304  Quadratic  Equations 

31.  The  sides  of  a  triangle  are  18  inches,  16  inches,  and  9 
inches.  By  how  much  may  the  sides  be  equally  shortened  so 
that  they  may  form  the  sides  of  a  right-angled  triangle  ? 

32.  Divide  a  straight  line  8  inches  long  into  two  segments 
such  that  double  the  square  on  one  segment  shall  equal  the  rec- 
tangle whose  base  and  altitude  are  respectively  the  whole  line 
and  the  other  segment. 

33.  Solve  the  equation  ax2  +  bx  +  c  =  0  when  a  =  5,  b  =  20, 
c  =  16. 

REVIEW  QUESTIONS 
427.    1.   What  is  a  quadratic  equation  ? 

2.  Illustrate  each  of  the  three  forms  of  quadratic  equations. 

3.  What  is  a  complete  quadratic  equation  ?  an  incomplete 
quadratic  ? 

4.  Give  the  rules  for  solving  incomplete  quadratics. 

5.  In  what  form  must  an  equation  be  written  if  it  is  to  be 
solved  by  factoring  ? 

6.  Give  at  sight  six  roots  of  the  equation  (x2  +  2x)(x2  —  1) 
(x2  —  5  x  +  6)  =  0.  Can  you  give  at  sight  any  roots  of 
(x2  +  2  x){x2  -  1) =  37  ?     (Explain.) 

7.  What  is  the  p-i orm  of  the  quadratic  equation  ?  How 
is  the  quadratic  in  one  unknown  reduced  to  the  p-i  orm  ?  Why 
is  the  j>form  used  when  solving  by  completing  the  square  ? 

8.  Reduce  7  x2  —  3  x  +  2  =  5(3  —  x)  to  the  p-form  and  give 
the  value  of  the  absolute  term  when  in  the  p-form. 

9.  Given  the  equation  x2  +  2  x  +  1  =  9.  In  solving  this 
equation  the   next   step   gives   x  -f  1  =  ±  3.     Why  is   it  not 

±(a  +  l)=±3? 

10.  Can  you  solve  a  quadratic  equation  that  lacks  the  abso- 
lute term  by  completing  the  square  ? 


XVIL    SIMULTANEOUS  EQUATIONS  INVOLVING 
QUADRATICS 

428.  One  equation  of  the  first  degree  and  the  other  of  the  second 
degree. 

1.  Of  what  degree  in  a;  is  ax2+  bx  +  c  =  0  ?  of  what  degree  in 
a  ?  (See  §§  245,  246.)  Of  what  degree  in  x  and  y  is  2  z+ y =10  ? 
Of  what  degree  in  x  and  y  is  3  xy  =  1  ? 

2.  What  is  the  principle  of  substitution  ?     (See  §  374.) 

3.  Explain  the  solution  of  simultaneous  equations  of  the  first 
degree  by  the  method  of  substitution. 

429.  Solve  the  simultaneous  quadratic  system, 

x  +  y  =  6, 
x2  +  3y=l6. 
y  =  6  —  x.     (From  the  first  equation.) 
X2  +  3(6  -  x)  =  16.     (Substituting.) 
x2  -  3  x  =  -  2. 
x2  -  3  x  +  f  =  I 

x  =  2  or  1. 

y  =4  or  5.     (Substituting  in  the  first  equation.) 
There  are  two  sets  of  answers,    x  =  2,  y  =  4  will  satisfy  both  equations. 
Also  x  =  1,  y  =  5  will  satisfy  both  equations. 

430.  To  solve  a  system  of  simultaneous  equations  when  one  equation 
is  of  the  first  degree  and  the  other  of  the  second  degree : 

1.  Find  the  value  of  one  of  the  unknown  numbers  in  terms  of  the 
other  unknown  and  known  numbers  from  the  first  degree  equation. 

2.  Substitute  the  value  of  the  unknown  thus  found  in  the  second  de- 
gree equation  and  solve  the  resulting  quadratic. 

3.  Substitute  each  value  of  the  unknown  already  found  in  the  original 
linear  equation  and  solve  for  the  other  unknown. 

4.  Arrange  the  answers  in  pairs  as  found. 

305 


306    Simultaneous  Equations  Involving  Quadratics 

EXERCISE 

431.  Solve  the  following  systems  of  simultaneous  quadratic 
equations.  Find  results  involving  decimals  correct  to  two  decimal 
places : 

1.  x  —  y  =  2,  (Why  is  it  better,  in  this  example,  to  substi- 
x2  -f  xy  =  40.       tute  *  —  2  than  to  use  x  =  y  +  2  ?) 

2.  2  x2  -  y2  =  7,  12.    15(a2  -  ?/2)  =  16  xy, 
2x  —  y  =  S.  x  —  y  —  2. 

3.  3  x  —  y  =  5,  13.   ic  -f  ?/  =  15, 
xy  —  x  =  0.  x2  +  y2  =  150. 

4.  %2  +  'u2  =  40,  14.  ^2  +  ?2  =  25, 

w  =  3v.  3p  +  4  g  =  24. 

5.  5  ic2  +  y  =  3  xy,  15.   x2-\-2xy  —  y2=7(x—y)} 
2x  —  y  —  0.  2x  —  y  =  5. 

6.  (z  +  y)(a;  —  2y)=7,  16.    r  :  s  =  9  :  4, 

a;  -  y  =  3.  r  :  12  =  12  :  s. 

7.  an/ =  135,  17     ^2  +  y  +  l=3 
aj  =  3  i/2  +  ^  +  l      2' 
2/5                                                 a:  -  ?/  =  1. 

8.  x2-y2  =  240,  18.   a?y  =  360, 
a?  —  y  =  6.  a?  —  y  =  9. 

9.  x  +  y  =  37,  19.    a=2&, 

a2  +  2/2  =  949.  (a  +6)2-(a2+62)  =  100. 

10.  m24-^2  =  130,  20     a!2H-2/2  =  a;  +  2/  +  2  =  5 
m  +  n:m  —  n  =  8 : 1.  #  +  3/  3  3 

11.  a;2  +  2/2  +  «2/ =  147, 
(B  +  2/  =  13. 

21.    —  4-  —  =  13,     (Regard  -  and  -  the  unknowns.) 
x2      y2  x        y 

i+l-* 

a;      v 


Simultaneous  Equations  Involving  Quadratics    307 

22.    2/-z  =  8,  1  +  1  =  1. 

2/2  =  240.  a     b     20' 


23. 


^  =  12.  i  +  i=41 


x  —  y 

x*-y*  =  48. 


a2     62     400 


432.  Many  of  the  problems  in  §  426  conld  have  been  solved 
by  using  two  unknown  numbers  instead  of  one.  In  general, 
the  student  will  find  it  easier  to  state  such  problems  alge- 
braically by  using  two  unknowns  than  by  using  one  unknown. 

433.  Problems  Involving  Simultaneous  Quadratics. 

1.  The  difference  of  two  numbers  is  4  and  the  sum  of  their 
squares  is  106.     Find  the  numbers. 

The  equations  required  are  evidently  x  —  y  =  4,    ar*-f  ?/2=106. 
Let  the  student  solve  the  system. 

2.  The  sum  of  two   sides   about   the  3* 
right  angle  in  a  right-angled  triangle  is 
17    inches,    and    the    hypotenuse   is   13  y 
inches  long.     Find  the  sides  about  the  right  angle. 

Solution.     Let  X  =  the  number  of  inches  in  one  of  the  sides, 
and  y  =  the  number  of  inches  in  the  other  side. 
Then  x  +  y  =  17,  (By  the  first  condition.) 
and  x2  -f  y2  =  169  (By  the  second  condition.) 
x  =  17  -  y. 
(17-y)2  +  y2  =  169. 

289  -  34  y  +  y*  +  y*  =  169.     (Why  ?) 
y2  _  17  y  =  _  60.     (Why?) 

yl  _  17  y  +  i|9  =  _4^.       (Why  ?) 

y-V  =  ±h 

.-.  y  =  12  or  5, 
and  x  =  5  or  12.     (Why  ?) 
Therefore  the  sides  about  the  right  angle  are  12  inches  and  5  inches. 

3.  The  sum  of  two  numbers  is  21  and  their  product  is  68. 
What  are  the  numbers  ? 

4.  The  perimeter  of  a  rectangle  is  27  feet  and  the  area  is 
44  square  feet.     What  are  the  dimensions  ? 


308    Simultaneous  Equations  Involving  Quadratics 


5.  The  perimeter  of  a  rectangle  is  34  inches,  and  the  diag- 
onal is  13  inches.     What  are  the  dimensions  ? 

6.  Two  fields  of  unequal  size  are  both  square.  Their  total 
area  is  50  acres  and  it  takes  1^  miles  of  fence  to  inclose  them. 
Find  the  dimensions  of  the  fields. 

7.  The  sum  of  the  areas  of  two  circles  is  13,273.26  square 
yards  and  the  sum  of  the  radii  is  79  yards.  Find  the  lengths 
of  the  radii. 

8.  The  product  of  the  sum  and  the  difference  of  two  num- 
bers is  a  and  the  quotient  of  the  sum  divided  by  the  difference 
is  b.     Find  the  two  numbers. 

9.  The  area  of  a  rectangle  is  1224  square  feet  and  the 
unequal  sides  are  in  the  ratio  of  3  to  5.     Find  the  dimensions. 

10.    A  line  AB,  10  inches   long,  is  divided   at  P  into  two 


y 


parts,  x  and  y,  so  that  a;  is  a  mean  proportional  between  AB 
and  y.     Find  the  lengths  of  x  and  y. 

11.  In  a  right-angled  triangle  the  hypotenuse  is  20  inches 
long  and  the  sum  of  the  other  sides 
is  28  inches.     Find  the  other  sides. 

12.    The  hypotenuse  of  a  right  tri- 
angle is  10  inches  and  the  perimeter  is 
24  inches.    Find  the  length  of  the  two  sides  about  the  right  angle. 

13.  The  area  of  a  right-angled  triangle  equals  one  half  the 
product  of  the  sides  about  the  right  angle.  If  the  area  of  a 
right-angled  triangle  is  30  square  inches  and  the  sum  of  the 
sides  about  the  right  angle  is  20  inches, 

find  the  length  of  these  sides  correct  to 
two  decimal  places. 

14.  The  perimeter  of  a  rectangle  is 
26  inches  and  one  of  the  diagonals  is  10 
inches  long.  Find  the  lengths  of  the  sides. 


General  Review  309 

GENERAL  REVIEW 

434.    1.   Factor  (a)  x2  -  6  ax  -  9  b2  -  18  ab. 
(6)  24  x2  +  6  ay  -  18  ?/2.     (Princeton.) 

2.  Find  the  L.  C.  M.  and  the  H.  C.  F.  of 

O3  +  a3)(»2  +  a2),    (x2  +  az  +  a2)(3  x  -  a),  3  x2  4-  2  aa  -  a2. 

(Harvard.) 

3.  Simplify  (_^t_iI)+(-lt--A_)>    (Regents.) 

4.  A  number  multiplied  by  17  is  increased  by  1056.  What 
is  the  number? 

5.  In  1912  a  father's  age  was  three  times  that  of  his  son 
who  was  born  in  1890.  When  will  the  son's  age  equal  one 
half  the  father's  age  ? 

6.  Solve  the  system  by  addition  or  subtraction : 

a-3^6     x  +  5  =  7 
y-4:      7'    y  +  1      6* 

7.  Solve  by  substitution  : 

9*  =  13y>    X-l  =  l- 
y     5     35 

8.  Divide  x*  -  3  x3  -  36  x2  -  71  x  +  &  by  x2  -  8  x  -  3. 

9.  For  what  value  of  A:  in  example  8  will  the  division  be 
exact  ? 

10.  A  and  B  start  from  the  same  place,  A  traveling  due 
north  and  B  due  west.  B  travels  one  mile  an  hour  faster 
than  A  and  at  the  end  of  3  hours  they  are  15  miles  apart. 
What  is  the  rate  of  each  ? 

11.  Resolve  into  factors  : 

(a)  ^-3^+2.         (6)  tf-y*.         (c)  9-6C  +  C2. 
y2         x2 

12.  Simplify  — 5 £±4  -  t^ — 

x+1      x-1      1-x2 

> x+  1         x2 

x  +  1      x  —  1      1  —  x2 


13.    Solve— L_-£±i —  =  0. 


310  General  Review 

12  3  1 

°YG  2x-l~x  +  2~ 2x  +  2     2^  +  Sx-2  = 

15.  Simplify    a?~4  _3a>-5     5*  +  9s  +  U 

*    J  2x-l       x  +  2       2x2  +  %x-2 

16.  Find  the  value  of  8  in  8  =  |  gt2  4-  vQt  when  t  =  3,  a  =  32, 
and  -y0  =  0. 

17.  Find  g  in  example  16  if  8  =  277.6,  t  =  4,  v0  ==  5. 

18.  Find  £  in  example  16,  if  8  =  450,  a  =  32,  v0  =  10. 

19.  Solve  3  x2  —  7  a;  —  2  =  0,  finding  the  values  of  x  correct 
to  two  decimal  figures. 

20.  Divide  x4  4-  a?  4-  ax2  +  6a;  —  3  by  a?  4-  2  a?  —  3,  and  find 
what  values  a  and  b  must  have  in  order  that  there  shall  be  no 
remainder. 

oi     rci™  «      c    __    2a  —  3  6      2c  — 3d 

21.  When  -  =  _,  prove  _^-  =  ^_. 

22.  Solve  5s-g(^  =  3^-5). 

a;       3  2 

23.  Find  VO  to  3  decimal  figures. 

24.  A  room  is  one  yard  longer  than  it  is  wide.  At  $  1.75 
a  square  yard  a  carpet  for  the  floor  costs  $  52.50.  Find  the 
dimensions  of  the  room. 

25.  Solve  ax  —  by  =  0,  x  —  y  m  c. 

26.  Factor  (a)  27  a? -64; 

(6)  16a-25a&2; 

(c)  16 a;2  4- 25 ?/2  4- 40 a#; 

(d)  a*4-2/6; 

(e)  a;  —  1  4-  x3  —  x2.  (Eegents.) 

27.  Simplify 

_b rf-y     /a  +  6     q-i\,/    a     ,      6    \ 

1_1  a»  +  b*     \a-b^a  +  b)     \a  +  b^a-b) 

b     a  (Regents.) 


General  Review  311 

28.  Find  the  square  root  of  5  x2  -  23  xA  +  12  x  +  8  xh  -  22  x* 
+  16  x6  +  4. 

29.  Solve  the  system  x+  y  +  z  =  4, 2x4-31/  —  z  =  1,  3  <c  —  y 
+  2^  =  1. 

30.  Factor  (a)  a2  -  4  ax  -  4  62  +  8  ab  ; 

(6)  (a  +  6)(c2  -  d2)  -  (a2  -  62)(c  -  d). 

-i        f.'+iY-i 

„.   Simplify  ^3^-^        V        ^ J.. 

x  +  1-2     (l-iY^-l+i) 

32.  Solve  the  system  x  +  y  +  z=l,  mx  +  y+z  —  0,  4#-f4y 
-32  =  0. 

33.  For  what  value  of  m  will  the  value  of  x  and  z  be  the 
same  in  example  32  ? 

34.  A  train  makes  a  run  of  120  miles.  A  second  train 
starts  one  hour  later  and  traveling  6  miles  an  hour  faster 
reaches  the  end  of  the  same  run  20  minutes  later  than  the  first 
train.     Find  the  time  of  the  run  for  each  train. 

35.  Solve  (x  -l)(x-2)=  15. 

36.  There  are  ten  numbers  in  a  series  as  follows :  x,  x  -+-  y7 
x  +  2y  —  x  +  9y.  The  product  of  the  first  and  last  is  70,  and 
the  sum  of  all  the  numbers  is  95.     Find  the  numbers. 

37.  A  chauffeur  engages  to  accomplish  a  journey  of  100 
miles  in  a  specified  time.  After  he  has  traveled  50  miles  at  a 
rate  that  will  just  enable  him  to  keep  his  engagement,  his  car 
is  delayed  20  minutes.  By  driving  the  remaining  distance  5 
miles  an  hour  faster,  he  reaches  his  destination  on  time. 
Find  the  original  rate.     (Sheffield.) 

1  1 


cx  =  0. 


38. 
39. 

Simplify  — 
Solve  3  bx  - 

x-2     3x  +  2 

9-i 

X2 

-7(x  +  b)+ac- 

312  General  Review 

40.  It  -  =  -,  prove  = • 

b     d'*  5  b2  5d2 

41.  If  v  ==  — - — -,  find  the  value  of  t  in  terms  of  the  other 

b  —  at 

letters.     (Princeton.) 

42.  Find  the  L.C.M.  of  6x2-5x-6,  3  +  x-2x2,  2x* 
-3x2-2x  +  3. 

43.  Factor  into  linear  factors  4  a2b2  —  (a2  +  b2  —  c2)2. 

(Princeton.) 

44.  Factor  (a)  32  a?bz  -  4  66 ;  (6)  #2  +  2  xy  -  a2  -  2  ay. 

45.  Solve  x  -  y  =  4,   ±— ±»JJL 

y      a;      117 

46.  How  much  water  must  be  added  to  80  pounds  of  a  5  % 
salt  solution  to  obtain  a  4  %  solution  ?     (Yale.) 

12  3 

47.  If  m= -,   n  =c -,  »= ;  find  the  value  of 

a  +  1  6  +  2  a+3' 

— ^-  +  — —  +  t^—     (Univ-  Qf  penn.) 
1  —  m      1  —  n      1  — p 

48.  Simplify" - (Harvard.) 

b2     £2         cd 

c 

a 

b 

49.  Evaluate  o  -  \5b-[a  -(3a-36)  +  2c-3(a-26  — o)]|j 
if  a  =  -  3,  b  =  4,  c  =  -  5.     (Yale.) 

50.  A  train  from  Chicago  to  Denver  running  at  an  average 
rate  of  40  miles  an  hour  makes  the  journey  in  6f  hours 
shorter  time  than  one  that  runs  32  miles  an  hour.  Find  the 
distance  from  Chicago  to  Denver. 

51.  The  rates  of  the  trains  remaining  as  in  the  last  problem, 
the  faster  of  two  trains  from  New  York  to  Chicago  makes  the 
run  in  6  hours  less  time  than  the  slower  train.  Find  the  dis- 
tance between  New  York  and  Chicago. 


General  Review  313 

52.  Find    by   factoring   the   H.  C.  F.   and    the   L.  C.  M.   of 

Xi  +  a2  -  b2  +  2  ax,   x2  -  a2  +  b2  +  2  6x  and  x2  -  a2  -  62  -  2  a&. 

(Harvard.) 

53.  A  and  B  each  shoot  30  arrows  at  a  target.  B  makes 
twice  as  many  hits  as  A  and  A  makes  three  times  as  many 
misses  as  B.     Find  the  number  of  hits  and  misses  of  each. 

(Univ.  of  California.) 

54.  I  have  $  6  in  dimes,  quarters,  and  half  dollars,  there 
being  33  coins  in  all.  The  number  of  dimes  and  quarters 
together  is  ten  times  the  number  of  half  dollars.  How  many 
coins  of  each  kind  are  there? 

55.  Write  by  inspection  all  the  roots  of 

(x2  +  2 x){&  -  3x  +  2)(x  -  10)  =  0. 

56.  Solve  x2  —  1.6  x  —  .23  =  0,  obtaining  the  values  of  the 
roots  correct  to  three  significant  figures.     (Harvard.) 

57.  Solve  ^±4  =  !^=Jj  -  1 ^ (Princeton.) 

3a  +  2      3z-2  4-9a?      v  ' 

58.  A  train  running  30  miles  an  hour  requires  21  minutes 
longer  to  go  a  certain  distance  than  does  a  train  running  36 
miles  an  hour.     What  is  the  distance  ?     (Cornell.) 

59.  A  physician  having  100  cubic  centimeters  of  a  6  %  solu- 
tion of  a  certain  medicine  wishes  to  dilute  it  to  a  31  %  solu- 
tion. How  much  water  must  he  add  ?  (A  6  %  solution 
contains  6  %  of  medicine  and  94  %  of  water.)     (Case.) 

60.  Solve  2x  +  5y=  85,  2y  +  5z  =  103,  2z  +  5y  =  57. 

(Vassar.) 

61.  Find  the  values  of  k  that  will  satisfy  the  equation 

fc2-a2-8fc-4a  +  12  =  0. 

62.  A  workman  receives  $  3.60  for  his  regular  day's  work 
and  double  pay  for  overtime.  In  a  certain  day  he  received 
$  5.20  for  11  hours'  work.  How  much  of  the  time  was  over- 
time? 


314 


General  Review 


63.  A  half  mile  race  track  is 
to  be  laid  out  with  semicircu- 
lar ends  in  a  rectangular  field. 
If  each  of  the  straight  sides 
is  484  feet  long,  what  must 
be  the  radius  of  the  semi- 
circular ends  of  the  track. 
(Use7r  =  3f) 

64.   The  horse  power  (H.  P.)  of  a  gasoline  engine  is  given 

D2N 


approximately  by  the  formula  H.  P.  = 


2.5 


where  D  is  the 


diameter  of  the  cylinder  in  inches  and  N  is  the  number  of 
cylinders.     State  this  formula  as  a  rule. 

65.  Using  the  formula  of  example  63,  find  the  H.  P.  of  a 
two  cylinder  motor  boat  engine  if  the  diameter  of  each 
cylinder  is  5  inches.  « 

66.  What  is  the  approximate  diameter  of  each  cylinder  of 
a  six  cylinder  40  H.  P.  automobile  engine  ? 

67.  Two  men  start  from  the  same 
corner  A,  going  in  the  directions  indi- 
cated around  a  field  1  mile  square.  The 
man  going  along  AB  walks  4  miles  an 
hour,  and  the  other  man  goes  3  miles 
an  hour.  Where  and  after  how  long 
will  they  meet? 


XVIIL    EXPONENTS 

435.  What  is  an  exponent  ?     (See  §  64.) 

Up  to  the  present  time  only  positive  integers  have  been 
used  as  exponents,  and  for  positive  integral  exponents  we  have 
developed  the  laws  for  multiplication  and  division. 

a™  .  an  =  om+n.     Multiplication  Law  (§  114). 
a™  -=-  an  =  a™-n.     Division  Law  (§  147). 

436.  Laws  of  Exponents  for  Involution. 

1.  To  find  a  power  of  a  power  : 

(a2)3  =  a2  •  a2  •  a2     (By  the  definition  of  exponent.) 

=  a6.     (By  the  law  of  exponents  in  multiplication.) 
.-.  (a2)*  =  a2x3. 
Also  (a4)*  =  aA-a*>a*  =  a12. 

.-.  (a*)»  =  a4*3. 
In  general,  (a™)*1  =  a™*1.     Power  of  a  Power. 

2.  To  find  a  power  of  a  product : 

(ab)3  =  ab-ab-ab     (Definition  of  Exponent.) 
=  a- a- a- &•&•&.  (§73.) 


Also         (abc)2  =  abc  •  abc  =  a-  a-b-b  •  c  •  c  =  a262c2. 
In  general,        (a&)n  =  anbn.     Power  of  a  Product. 

An  integral  exponent  of  a  product  can  be  distributed  to  the  factors  of 
the  product. 

315 


316  Exponents     » 

3.  To  find  a  power  of  a  quotient : 

(ay     «  .  a  §  a=a?t     (why9) 
\bj      b     b     b      63      ^vv   r'J 

In  general,        (  ~ )    =^.    Power  of  a  Quotient. 
\bj       bn 

An  integral  exponent  of  an  indicated  division  can  be  distributed  to  the 
dividend  and  the  divisor. 

ORAL  EXERCISE 

437.  Perform  the  operations  indicated  : 

1.  a5  •  a4.  12.    rap+2  •  mp_3.  23.    (c5)4. 

2.  a;10  •  a7.  13.    t2v+1  •  p-»  24.    (ic7)9. 

3.  yt-tf.  14.   a10-^-6. 

4.  m2  •  ra°.  15.   a11  -r-  a7. 

5.  &c  •  b\  16.    m5  -r-  m2. 

6.  a2n-an>a.  17.   c* -^- c2. 

8.  ct*a.  19.    6c45-f-64.  '    Uv 

9.  cn_2c2.  20.   2/2x+1  -r-  2/*_1.  28.    (xnypy. 

10.  d**1.^-1.  21.    a2n"3-^an+1.  29.    (t""1  •  sn"2)w. 

11.  aj^+V"1.  22.    (a2)5.     ■  30.   (a263)4. 

438.  According  to  the  definition  of  an  exponent  (§64),  such 
expressions  as  a~*,  a0,  a~5  have  no  meaning,  since  it  is  impossible 
to  use  a  two  thirds  times,  or  zero  times,  or  —  5  times,  as  a  factor. 
It  is  convenient,  however,  to  use  fractions,  zero,  and  negative 
numbers  as  exponents  and  to  define  them  in  such  a  way  that 
the  laws  for  positive  integral  exponents  shall  hold  for  these 
exponents. 

439.  Fractional  Exponents.  Assuming  the  law  of  multiplica- 
tion to  hold  for  fractional  exponents,  we  have  a*  >  a?  =  a  *+2  =  a. 

,\  a*  =  Va,  since  a2  is  one  of  the  two  equal  factors  of  a. 


25. 


26. 


{-ft- 


Exponents  317 


Similarly,  oft  •  eft  •  a?  =  a*"1^**  =  a. 

.-.  a*=^a.     (Why?) 
l 
In  general,  a**  =  Va. 

Again,  a*  >  a*  -  eft  =  eft****  =  a2. 

.-.  a*=%l  (Why?) 
Similarly,    a*  .  a*  •  a*  •  a  I , a*  -  «*W+I+W  =  as. 

.-.<**  =#5  (Why?) 
In  general,  ai  =  a/op. 


440.  Stated  in  words  we  have  : 

The  numerator  of  a  fractional  exponent  indicates  a  power  and  the  de- 
nominator indicates  a  root. 

1       q 

Thus,  2a*  =  2#a»;  3o'6S  =  S^aV6?;  8$  =  y/¥  or  (#8)*  =  32. 

EXERCISE 

441.  Write  with  radical  signs,  noting  carefully  ivhat  is  the  base 
for  each  fractional  exponent  : 

a*.  3  r«. 


\Sx) 


ft. 

6.    a  + 


2-    a  •  .     _  .  6i  10.    5r<s2. 


3-  2a;-  7.  (a +  6)1  11.  5z2«)<. 

4.  (3a)hi  8.  a +2  6*.  12.  2  a* +3  6*. 
Write  with  fractional  exponents  : 

13.  Va2.  17.  va  +  J.  21.  Va2  -  62. 

14.  Va*.  18.  a+V&.  22.  2Va+V2a. 

15.  3-s/a*.  19.  4^(a  +  6)2.  23.  Va +Va. 

16.  V2a6.  20.  3Vx-VaF.  24.  H/S?  +  V^. 


318  Exponents 

Find  the  values  of  the  following : 

25.  16*.      (Extract  the  root  first.) 

26.  42;  8*.  31.  £-25*. 

27.  27'  ;  9l  32.  25*  +  216*. 

28.  (-125)1  33.  243* -256*. 

29.  27^-27^.  34.  (-8)*.  (-8)*. 

30.  9*.  9*.  35.  18769*. 

36.  1.5*  to  two  decimal  places. 

37.  Apply  the  third  law  of  exponents  to  (aJ)3.     What  does 
the  result  suggest  as  to  the  meaning  of  a$  ? 

442.  Zero  Exponent.     If  we  assume  law  1  to  hold  when  n  =  0, 

we  shall  have : 

am  'd°  =  am+0  =  am, 

or  am  •  a0  =  am. 

Dividing  by  am,  aP  =  am  -*-  am  —  1. 

.-.  fl°  =  l. 

443.  Stated  in  words  we  have  : 

Any  base  with  the  exponent  zero  is  equal  to  unity. 
Thus,  a0  =  20  =  100«  =  (x  +  y)°  =  1. 

444.  Negative  Exponent.     If  we  assume  law  1  to  hold  when  n 
is  a  negative  number,  we  may  write  : 

a"3  •  a3  =  a"3+3  =  a0  =  1, 
or  a~3  •  a3  =  1. 

Dividing  by  a8,  a-3  =  —  • 

a3 

In  general,  a~n  •  an  =  a~n+n  =  a0  =  1, 

or  a~n  •  an  =  1. 

Dividing  by  an,  a~M  =  — 


Exponents  319 

445.  The  last  equation  is  the  definition  of  a  negative  expo- 
nent in  algebraic  symbols.     Stated  in  words  we  have  : 

Any  base  affected  by  a  negative  exponent  is  equal  to  1  divided  by  that 
base  with  a  positive  exponent  of  the  same  absolute  value. 

Examples 

1.  3^  =  3-1  =  3.  3.    (xyy  =  l. 

2.  Sx-l  =  S.-  =  --  4.    a°  +  a*=l  +  l  =  2. 

x     x 

a2     b2        aW 

ORAL   EXERCISE 

446.  Simplify  by  using  the  definitions  of  exponents  and  reducing 
the  results  when  possible : 

1.  a*-a«.  12.    (-i)"3;   (-^-2. 

2.  4(a  +  6)°.  13.    92. 3"6. 

.    3.    [4(a  +  6)]°.  14.   3-2.2-3;  (-3)~2;  -3"2. 

4.   4(a° +  &<>).  15.    V25-5-1. 

5-  («°)n-  16.  (a0  +  b°  +  c°)-\ 

6.  1».(-1)3.  17#  64.2-6. 

7.  (x-yy>.5-\  18#  4-2_j_2"4. 

8.  (25i  +  8*)-1-  19-  64-1-26. 


9.    3  a"1;   S^a.  20.    a*  -  or1  -  or2. 

0.   9*.  3"2;   9  +3-2. 
11.    9-*-3"2;   9-!  +  3- 


10.   9.3-;   9+3-2.  ^   a0+I 

a0 


EXERCISE 
447.   Simplify  as  much  as  possible : 

1.  (25*  +  8*)-2-  4'   9'4-5_1- 

2.  8-2-321  5.   5-2+(-i)3. 

3.  22  +  2-i.4i  6.   25-5-1. 


320  Exponents 

Simplify  as  much  as  possible  : 

7.  a3-K-a)2;  (-ay  +  a2.  11.  3  .  3"1  +  4  -^  4rK 

8.  100*  +  100*  +  100-i  12.  (2«  +  30)3;   (23  +  33)0. 

9.  3-2  +  27*  13.  -^8.  8^;   2~2  +  8~l 
10.   3"2  +  3.9l  14.  16* +  2-2.  8*. 

448.     Negative  Exponents  in  Fractions. 

ab'1     a    7_i      a    1      a       /tt.     i   •  i     .      \ 

1.    =  -  •  b  l  =  - .  -  =  —     (Explain  each  step.) 

c        c  c   b     be 

ab^ab  ;1     *£;     (Explain.) 
2c      2     c         2  v     r        ' 

3.    -=al>b- =  -•& — d  = — 

cd~x  c   d~l     a        c  ac 

A     /a\-2        1         1       b2     fbV      «     ,.    v 

4.  f  _  ]     = =  —  =  —  =[-]  •     (Explain.) 

\bj         fa\2     a2     a2      [a         K     ^        ' 


a\2      a2      a2 
V       62 


449.  These  examples  illustrate  the  following  principles  :     . 

1.  Any  factor  of  the  numerator  of  a  fraction  may  be  made  a  factor  of 
the  denominator,  or  any  factor  of  the  denominator  may  be  made  a  factor 
of  the  numerator,  if  the  sign  of  its  exponent  is  changed. 

2.  Any  fraction  affected  by  a  negative  exponent  is  equal  to  the  re- 
ciprocal of  that  fraction  with  the  sign  of  its  exponent  changed. 

The  student  should  carefully  note  that  factors,  not  terms,  can  be  changed 
from  the  numerator  to  the  denominator  or  from  the  denominator  to  the 
numerator  by  changing  the  sign  of  the  exponent. 

EXERCISE 

450.  In  the  following  examples  make  the  exponents  positive 
and  simplify  the  expressions  as  much  as  possible : 

1.   2  a;"3.  4.    a-lb~x. 

2-   2a~'x'  5.   a-i  +  b-\ 

3     -fL+°L. 

x~A     x-*  6.    (a  +  6)"1. 


Exponents  321 


7     a-l  +  b'\ 

11. 

(«)-'• 

x-1  +  y1 

8.  (-3)3.  3-3. 

9.  a-  +  h-. 
b2     a~2 

12. 
13. 

(-i)-3- 

10-  aJ£ 

arAb 

14. 

2a-16"1 
a~2  -  6-2 

Write  examples  15 

to 

26 

in  in 

tegral  j 

brm,  using  negative  expo- 

nents  when  Jiecessary 

15.    1. 

a 

19. 

X 

23.    i-i- 

X-     y2 

,.     1 

# 

1 

16.    — •  20.    -=—  •  24. 

a2  —  z3  x  -f  ?/ 

„.    (!Y.  21.    *L.  25.    ttl. 

\a2J  xy2  x  —  y 

18.    «1.  22.      1  .  26.    £i±£. 

a_m  2_1  #*/ 

Find  the  numerical  values  in  examples  27  to  34  : 

27.    !._*_.  29.    A.  2-  32'    48'10-5- 

50    C_4)-3  3-2 

o.     1    *  1  33.   5  +  17  •  10-3. 

28      X     2-4  3°-    100,5    ' 

Mm    4-2        *  31.    1000-5-2.  34.    2135- 10-7. 

35.    Write  with  positive  exponents  and  simplify  the  result : 

a"3  +  b~3 
a-2-6-2' 

Solution.     _1_     J_     ft8  -f  a3 

a3     63        a363        6s  +  a3        a262 


1       1  ~  &2  _  CT2  ~    a8&3        62-a2 
a2     62       a262 

_  b2  -ba  +  a2 
ab(b  —  a) 


322  Exponents 

Write  examples  36  to  43  with  positive  exponents,  and  simplify 
the  results : 

36.    "-""' 


a  +  a-1 
a~3b~m 


•  (ir+& 


38. 
39. 


x-*y~n 
3  a-1 
3+ a"1' 


40. 

2  a;"1  -  6 
2a- 6"1 

41. 

a"1  +  6"1 
a~2  +  6-2 

42. 

^•(* +«•>-• 

43. 

2(a  +  6)-1  +  2(a-6)-1. 

451.  We  shall  assume  that  all  the  laws  of  exponents  that 
have  been  established  for  positive  integral  exponents  hold  for 
the  other  exponents  that  have  been  defined.  For  convenience, 
we  repeat  here  the  four  definitions  and  the  five  laws  of  ex- 
ponents in  algebraic  symbols. 

Definitions  of  Exponents 

1.  an  =  a  •  a  -  a  •••  to  n  factors  when  n  is  a  positive  integer. 

p 

2.  aq  =  yaP.    Fractional  Exponent. 

3.  a0  =  1.     Zero  Exponent. 

4.  a-w  = Negative  Exponent. 

Laws  of  Exponents 

1.  a™  >an  -  arn+n.     Multiplication  Law. 

2.  a1*1  -=-  an  =  a™-*1.     Division  Law. 

3.  (arn)n  s=  anin.    Power  of  a  Power. 

4.  (ab)n  =  anbn.    Power  of  a  Product. 

Power  of  a  Quotient. 


\b)         bn 


The  definitions  of  exponents   and  the  laws  of  exponents 
should  be  thoroughly  committed  to  memory. 


Exponents  323 


ORAL  EXERCISE 
452.    Apply  law  1  to  each  of  the  following  : 
1.  a^a~\  11.    («+&)(«  +  b)~ 


2.  a5  •  a"5.  12.    (a  +  b)\a  +  b)~\ 

3.  ym  •  y°.  13.   (a  -f  b)~\a  +  6)°. 

4.  xm-x~n.  14.    (a;  +  ?/)n+1(a; -f  y)2-«. 

5.  rm  •  r.  15.   5  aic-6  •  5~2abx7. 

6.  rm+3  •  rm-3.  16.    (a  —  »)"3(a;  —  a)"2. 

7.  6n+2  •  62~n.  17-    (a  ~  x)~4(v  —  a)6. 

8.  a""1  •  a  •  a\  18-    ^/a^-y/a-  aK 

9.  an_1  •  an_1.  19.    Va~x  *  (T*, 
10.  a"3  •  6~2  •  65  •  a3.                          20.   d*  .  d^Vd. 

Appty  law  %  t0  examples  21  to  41 : 

21.  a8  +  a\  32.    3a°6"  -s-(—  &»-2). 

22.  a8  ^-  a"3.  33.    (a;  -  y)"1  ■+  (x  -  y)~\ 

23.  a~h+ah.  34.    (X-yf^{y-x). 

24.  a5  -f-  a-5. 

25.  bn~z  +  b\ 

26.  or2&3  -5-  cr362. 

27.  a~3 '--(-a)-5. 

28.  or3-=-(-  a2). 

29.  a"3^-(-  a2). 

30.  2n~3 -j- 2""4. 

31.  2arn^an-2. 

Apply  law  3  to  examples  42  £o  47  : 

42.  (a"2)"3;  (-a3)2.  45.    (a4)<>;  (a*>)-i. 

43.  (-a2)3;  (a2)"3.  46.    (-  &-*)-«;  (- 6"2)- 

44.  (a~3)-4;  (a3)"4.  47.    (*-"-•)*-*. 


35. 

2               -1 

a?  s-a  s. 

36. 

a^  -j-  a$. 

37. 

a-f-2a~?< 

38. 

(2x)o-r-(2a)_i 

39. 

Scrfo^-a^c0. 

40. 

(a  +  6)^_(a  +  6)-i. 

41. 

25a"t-5-iai 

324  Exponents 

Apply  law  3  to  examples  48  to  61 : 

48.  (ax+v)x~v.  55.  (m*-v)x+v. 

49.  [(2a)"3]-i.  56.  Vxyz+(xyz)t. 

50.  [(x  +  y)-^.  57-  C(-«)4]3- 

51.  (^-i)«+i.  58*  [(«"")"'?• 

52.  (2e)f;[(-2)-]-t.  59.  [(~f)J 

53.  (a2)";  (a»)3.  60.  (a-i).    ' 

54.  {xn~iy.  61.  (2~2)-3. 

Apply  laws  4  and  5  in  ^e  following : 

62-   (2a<6°)'.  72^  3M-.    6 


63.  (-a&)2;  (a.&)n~2.                                      VV       a 

64.  (-a&)3;  (-a&)4;  (-a&)«.  73.  [(x  -  y)(y  -  xyj. 

65.  (#•#)".  74.  [(a;  -  y)\y  -  a?)"2]3. 

66.  (3a&«-2)3.  75.  (i)-3(|). 

67  /flW\»  76.  [2(a>-2,)-i]-2. 

'  V2c/.  77  /          81           U 

/*.y  V«2  +  2a6  +  6V* 


^2 

68. 


69.    (^X\ 


78.       /«2+L2a6  +  62 


4 

79.    (6°  •  25*  •  a4)*. 
70.    [(x-y)(x  +  y)-*]-+.  80.   (8"i  •  a"366)-*. 


71.  f?r 

\xj 


'  *  w  '  '  V  *   / 


EXERCISE 
453.    1.    State  the  four  definitions  of  exponents  in  words. 
2.    State  the  five  laws  of  exponents  in  words. 

In  the  following  examples  use  definitions  and  laws  to  simplify : 

3.  9°  •  9-i  •  9*  •  9"*.  5.    (-64)"*. 

4.  (9"2)-i.  6.    (64)"*. 


7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 

18. 

19. 

20. 
21. 
22. 


.008-2 
5  •  2~2. 

7.7-1;  7  +  7-i 

•I"2;  (--I)2- 

(a°+&°)2;  (aPbo)\ 
(a2  +  62)o. 
4,  (a -by. 
tt)-<-16<> 
(-l)o-f(-l)2. 
(-l)o+(-l)2+(-l)3 
2 

9(a0+&0_f_c0)-2- 
_3 

a  *  -=-  2  a~l. 


Exponents 

26.    (a-i  +  fc-^-j-afc. 
a;    ^ajj 


325 


ic5  -r-  a?-5. 


23.    al 


24. 
25. 


27.    - 


28.  a-!-j-a°. 

29.  &8h-&8"*. 

30.  g"^~1. 

ambn 

31.  (o*6*)i 

32.  92  •  3-*. 

33.  (-z)3-=-(-a;)2. 

34.  [(i)-3]->. 

35.  1-4-2. 

36.  l-r-(-3)"3. 

37.  .2-1;  .5"2;  1.5-2. 

38.  5-2-i;  8-10-5. 

39.  (a**)-*. 

1  1 


40. 


41. 


(_3)-2'   (-3) 


-3 


xn~*  +  x\  42.    (a"2)-3;  (a"3)2;  (er3)- 

43.  (-a2)"5;  (-a5)"2;  (-a"5)"2. 

44.  (-a3)"4;  (-«~3)4;  (-a-4)-3. 

45.  (-a3)"2";  (-a2«)-3;  (_  <*-*»)-• 


46.    (-a2n"i)2. 

-2 


48. 
49. 


47. 


erW\- 
x~y) 


a46~6 
a%-8 


\a%-2 


50.  (a*  +  &*)  (a*  -  &i). 

51.  (a*  +  ^)2. 


326  Exponents 

In  the  following  examples  use  definitions  and  laws  to  simplify : 

52.  (a*  +  &"*)  (at  -  aV*  +  &""*> 

53.  Va  — 2aM  +  6. 

54.  (a*  4-  a"^)2  -  (a^  -  a-*)2. 

55.  (a*  -  2)  (a*  -  3). 

56.  (a  —  x)(a2  —  x2)~l. 
58.   Vcriar1*  =(afar*)*,  etc 

68.  -\/xfyx. 

69.  2Vi*Va*. 

70.  2  •  4*  •  8"1 . 

71.  3-^3. 

72.  V3-a/3. 

73.  Simplify      lt 2.3. 4. 5.6. 7    (ava?)6(- a?Vo)7. 

74.  Simplify 

( V2^)6  +  5  •  ( V2^/)5( -  V2^)  +  10 ( V2^)4(-  V2xy)2. 

76.  Show  that  2n+l  -  2n  =  2\     (Factor.) 

77.  Show  that  2"  +2n+2  =  5-2". 

78.  Show  that  5"  +  5"+1  =  6-5-. 

79.  Is  3-32  =  92?  Is  3  .3n  =  9B? 

80.  Show  that  2"  •  4*  =  23\  81.   Show  that  10  +  2~*  =  3. 

1°  -  2"1 

82.  (a0  +  or1  +  or2)  a2. 

83.  Compare  the  value  23'  with  (23)4  without  actually  per- 
forming the  indicated  operations. 


57. 

(a2&""M)-u. 

six? 

<?' 

s/x^y'2. 

(Vx^r2)2. 

63. 

58.    V 

59. 

VVVK 

60. 
61. 

64. 
65. 
66. 

\y/i/cr*. 
■fyaWtr* 

V(-  i-yxy. 

62. 

Vva. 

67. 

V(-Bvy, 

Exponents  327 

45J.  The  fundamental  operations  are  performed  upon  ex- 
pressions involving  fractional,  zero,  and  negative  exponents 
in  the  same  way  as  when  the  exponents  are  positive  integers. 

1.  Multiply  3  x  +  ar1  +  2  by  3  x  +  x~x  -  2. 
3X  +  2  +  X'1 

3  a?  —  2  4-  ar  Arranging  in  descending  powers  of 

9  X2  -+-  6  x  +  3  x,  the  absolute  term  being  considered 

_  (J  x  —  4  —  2  ar1  as  having  x°  for  its  literal  part,  we  have 

3  4-  2  or1  4-  ar2  tne  work  M  indicated  at  the  left. 
9  a,-2            -f-2              4-ar2 

2.  Divide  or1  +  8  by  aT%  —  2 a~*  4-  4. 
a"1 +  8  a-S_2a-3  +  4 


a"1- 

-  2  a~*  +  4  a~* 

2a"^-4a"H8 
2o'*-4a~*+8 

a"* -f-2 


3.    Find  the  square  root  of  x?  —  4  #%  5  +  4  ?/  *. 
a;2  —  4  a?^~^  +  4 1/-1 1  a;*  —  2  ?/* 


a; 
2a£-2y"* 


—  4a;*;y  2_|_4^-i 

—  4  a;*?/"2  +  4  y-1 


In  arranging  in  descending  powers  of  a  letter,  where  should  the  term 
not  containing  the  letter  of  arrangment  stand  ?  Arrange  x~"2  +  x2  +  2  in 
descending  powers  of  x. 

EXERCISE- 
455.    Perform  the  operations  indicated: 

1.  (a?2  +  x$  +  t)(«*  —  a*  +  1). 

2.  (a%~2  4- 1  4-  aryXa^r1  —  a?"V). 

3.  (a^  +  ar^a-ar3). 


328  Exponents 

Perform  the  operations  indicated : 

4.  (4a$-6a*  +  9)(2a*+3). 

5.  (2*-2*  +  l)(2*  +  l). 

6.  («i -a*)1. 

7.  (x-1  -  y-1  +  z-1)2  -  (x-1  +  2/-1  -  z~ly. 

9.  Arrange    in    descending    powers   of    x    and    multiply 
(—  x°  +  x%  +  af*)(aj°  +  a^  +  »~  *). 

10.  Arrange  and  multiply  (x~*  +  1—2  x~2)(x~2  +  a;-4). 

11.  Arrange  and  divide  8  a  +  a-2  +  6  a-1 4- 12  by  2  a-1  +  a~2. 

12.  (y-4-7r-2-30)-r-(V-2  +  3). 

13.  (a."2 -35 -2  a-1) -(a"1 -7). 

14.  (30  cr1  -  53  a~*  +  8) -*- (6  a"-*--l). 

15.  (3«-10**  +  3)+'(3a>*--l). 

16.  (1  +  8  a"1  +  15  a"2)  -*-  (1  +  3  a"1). 

17.  (a*  +  12  at  -  48  +  52  a*  - 17  a)  ■*■  (a*  -  2  +  a*). 

18.  (m*  -  36  -  21  m"^  -  3  m*-71  t»-*)+(l-3  m^-8  m"*). 

19.  (a;  +  a^  +  l)-5-(a^ -a;*  +  l). 

20.  (a"1  +  27)--(a~t-3a"3+9). 

21.  (a?-*  +  y"t)  -s-  (art  +  y~*). 

22.  Find  the  square  root  of  a}  +  12  a*  +  36. 


23.  Var2  —  2  x~ly  +  ?/2. 

24.  (a;-1  -  22  x^y~*  +  121  fty. 

25.  V2  +  a*ar2  +  a~*x*. 

26.  (a?"4  -  6  a;-3  +  13  x~2  -  12  or1  +  4)*. 

27.  (2a  +  2ar1  +  3  +  a*  +  ar*)K 

28.  (cr* +  6  a-1 +9)~*. 


Exponents  329 

Expand  by  applying  type  forms,  examples  29  to  36  : 

29.  (0*-&*)(a*+6*).  33.    (a*  +  2  a*)2. 

30.  (x$-y~ty.  34.   (a;* +  »"*)(#*— ar*). 

31.  (a^-affy.  35.   (a*  -  2) (a*  +  7). 

32.  (a*-&ty.  36.    (a"1  +  3)(a-1-5). 

37.  Factor  a  -  3  a*  +  2  ;  2ar2  +  33T1  -  2. 

38.  Factor  a  —  b  into  two  factors,  one  of  which  is  a2  —  6*. 
Hint,     a  -  b  =  (a*)2  -  (6^)2. 

39.  Factor  a  —  6  into  two  factors,  one  of  which  is  a*  —  b3. 

40.  Factor  (a)  x  -  20  &*  +  100. 

(6)  a  —  4  Va  —  5. 
(c)  a2c^  +  ac  +  aM  +  a^c2. 
(<£)#  —  aM  —  62x2  +  a^fti. 
(e)  2a  +  5a^3. 

41.  a*  •  of?  •  v^v^  •  a"1.  45.    (4)2p  •  (-f)2p  •  (|)2p. 

42.  f^Yi(?]~3fS\~A.  Hint.     42*  =  2*p.     (Why?) 

43.  (a2&')-^(a-3&-*)3.  ^    ^—j    +(_J       . 

44.  (a-2-6-2)-^---J.  47    2«(2»-1)»-s-(2"+1.2»-1). 

48.  Show  that  (2»+4  -  2.2"+1)  .  2  -*-«  =  3. 

49.  (xyz)x+v+z  -^r(xy+xyx+z^+y)' 

50.  (^  +  e-)2-2.  Y-3^2 

.KinA  55-    W*^"*)"1- 


L  wJ  ■  56-  (<rwer*d*)"*- 

\&~)  \&~)     '  58.   2  •  5"4  •  5-34-1  •  10°. 


330  Exponents 

Perform  the  operations  indicated : 

60.    (x  +  y)i.(x+y)L  '   [\  cd* )   J 

62.   What  is    the    sign    of    the    answer    in   (— 3)2m?     in 
(—  3)2m+1  ?     (In  each  case  m  is  any  integer.) 

Q3     27(*>  +  y°  +  gP)-»>  W^VV 


7(27a^r7a 

"V  4(o°  +  g 


67. 

2 


64. 


(9arV<r6)-t  \  4(o°  +  ?/0) 


85.    (^Y^JSLIV  69       „-.6-i 


4a<^g\4<  yo    a-i  +  2  6-i 


71.  Divide  it-"6  —  3  x~*y~2  +  3  «"2?/-4  -  2T6  by  a;"2  —  y~\ 

72.  Divide  a;"3  -f  x~2y~l  +  x~ly~2  +  y~z  by  ar1  +  2/_1. 

73.  Find   the   value   of    Va2  —  b2  when   a  =  x*  +  »~^   and 
b  =  as*  —  «T». 

74.  Show  that  ^22(^3)3(6_^)29|35  =(|)i 


XIX.  RADICALS 
CLASSIFICATION  OF  NUMBERS  AND  DEFINITIONS 

456.  Real  and  Imaginary  Numbers.  The  numbers  of  algebra 
are  divided  into  two  classes,  real  and  imaginary. 

Real  Numbers.  Real  numbers  include  all  positive  and  nega- 
tive integers,  positive  'and  negative  fractions,  and  all  indicated 
roots  except  even  roots  of  negative  numbers. 

Imaginary  Numbers.  Even  roots  of  negative  numbers  are 
imaginary  numbers. 

Thus,  5,  —  7,  £,  v/—  3,  V  5  —  V23  are  real  numbers. 
V—  3,   "V5-  V28  are  imaginary  numbers. 

457.  Rational  and  Irrational  Numbers.  Real  numbers  are 
divided  into  two  classes,  rational  and  irrational  numbers. 

Any  integer,  or  a  number  that  can  be  expressed  as  the  quo- 
tient of  two  integers,  is  a  rational  number. 

Thus,  3,  .25,  5|,  .333  •••  (=  i),  >/9,  y/—  27  are  rational  numbers. 

All  other  real  numbers  are  irrational  numbers. 

Thus,  V5,  V 9  +  V4  are  irrational. 

The  irrational  numbers,  so  far  as  we  shall  be  concerned  with 
them  in  elementary  algebra,  are  indicated  roots  that  can  be 
obtained  only  approximately. 

458.  Radical.     The  indicated  root  of  a  number  is  a  radical. 

Thus,  V2,  V9,  y/a  +  6,  V^i  are  radicals. 

331 


332  Radicals 

A  radical  may  be  a  rational  number,  an  irrational  number, 
or  an  imaginary  number. 

Thus,  a/4  is  rational,  V5  is  irrational,  and  V^~4  is  imaginary. 

Radical  Expression.  An  expression  that  contains  a  radical  is 
a  radical  expression. 

Thus,  Va,  3  +  V2,  (a  -f  Vb)2  are  radical  expressions. 

459.  Order  of  Radicals.  Indicated  square  roots  are  radicals 
of  the  second  order;  indicated  cube  roots  are  radicals  of  the 
third  order,  etc. 

Thus,  V3  is  of  the  second  order,  Va  is  of  the  third  order,  y/a2  +  b3  is 
of  the  fourth  order. 

460.  Index  of  a  Radical.  The  index  of  a  radical  is  the  num- 
ber placed  to  the  left  and  above  the  radical  sign  to  indicate 
the  order  of  the  radical.     The  index  of  a  square  root  is  omitted. 

Radicand.  The  expression  under  the  radical  sign  is  the 
radicand. 

Thus,  in  2v/3a36,  3  cfib  is  the  radicand,  and  5  is  the  index  of  the 
radical. 

461.  Surd.  An  irrational  number  which  is  the  indicated 
root  of  a  rational  number  is  a  surd. 


Thus,  V2,  Va  are  surds,  but  V9  and  V 2  +  V3  are  not  surds. 

Quadratic  Surd.  A  surd  of  the  second  order  is  a  quadratic 
surd. 

462.  Principal  Root.  It  has  been  seen  that  VI  =  ±  2. 
From  this  we  should  infer  that  x  -f-  V4  =  x  ±  2.  However,  in 
dealing  with  radicals  and  expressions  containing  radicals  it  is 
customary  to  use  only  the  positive  root. 

Thus,  x  +  VI  =  x  +  2,  and  x  -  V4  =  x  -  2. 

The  positive  square  root  of  a  number  is  its  principal  square 
root. 


Reduction  of  Radicals  333 

463.    Principle  1.     The  square  root  of  a  product  equals  the  product 
of  the  square  roots  of  its  factors. 

In  symbols,  Vab  =  VaVb. 

This  principle  follows  immediately  from  the  fourth  law  of 
exponents. 

Thus,  Vab  =  (ab)$  =  ah^  =  Va  •  Vb. 

Principle  2.    The  square  root  of  the  quotient  of  two  numbers  equals 
the  quotient  of  their  square  roots. 

In  symbols,  a/-=— =• 

**>     Vb 

This  follows  from  the  fifth  law  of  exponents. 


'a\i  _a%  _  Va 
b\      Vb 


Thus'VR? 

These  two  principles  may  be  stated  for  any  root. 


V*     Vb 


REDUCTION  OF  RADICALS 

464.  Case  I.     To  remove  a  factor  from  under  the  radical  sign. 

Vo2&  =  Va2  Vb  =  a  Vb.     (§  463) 
ytfb  =  VtfVb=aVb.      (Why?) 

465.  If  any  factor  of  the  radicand  is  a  perfect  power  of  the  same  de- 
gree as  the  radical  index,  it  may  be  removed  from  under  the  radical  sign 
by  extracting  the  required  root  of  the  factor  and  multiplying  the  result 
by  the  coefficient  of  the  radical. 

Examples 

1.  6^54  =6^27^2  =  18^2. 

2.  2 V2W  =  2 V(a462)2  ab  =  2  a2bV2ab. 


3.    V(a2  -  b2)(a  +  b)  =  V(a  +  b)\a  -  6)  =  (a  +  6)  Va  -  b. 


334  Radicals 


466 

sign : 

.    Whenever  possible, 

VIS.                         3. 

V4  ab\                   4. 

EXERCISE 

remove  factors  from  under  the  radical 

1. 
2. 

V9a4&2c 
Va3^3. 

19. 

20. 

21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 

5.  Va3  +  63. 

6.  fV27  65. 

7. 

Va2  +  a2b\ 

Va6  +  n. 

8. 

V5x3-20a2  +  20a;. 
-\/6a2&2. 

aVaM>. 

aVa^~\ 

V2(a3-3a26+3a&2-&3). 

9. 

v/(s  -  2/)TO. 

10. 

V(a  -  6)2(a2  -  62). 

11. 
12. 

V(o3  -  63)(a  -  6)2. 

Vm2*+1. 

13. 

2V(a2-&2)(a-6). 
Va2  -  b\ 

&Va668+°. 

14. 

V(#2  —  2/2)3. 

15. 

Vm2  —  2  mw  +  w2' 

Va-2"+1. 

16. 

VC^-S^  +  ^aj-lX2. 

17. 

V(a>  -  2/)2"1. 

Va4&4+*. 

18. 

Vm3  —  n3)(m  —  n). 

Va4  +  b\ 

When  the  radicand  is  negative  and  the  root  index  is  an  odd 
number,  the  negative  sign  should  always  be  removed  from 
under  the  radical  sign. 

Thus,    ^"16=^-  8  •  2=-2v/2;    V^a  =  </(-  l)3a  =  -  1  Va    or 
31.    -v/^32.  34.    A/-m6n.  37.    ^^160. 


32.    5a-</-a\        35.    V-a6-a7.  38.    V-500. 

3 


33.    2x  +  lJ  -tf.         36.    Va3-«6-  39.    V-  m9n7. 

40.    ^12^+^87.       41.    S^4--^"^!. 


Reduction  of  Radicals  335 

467.  Case  II.  To  change  a  radical  whose  radicand  is  a  fraction 
to  an  equivalent  radical  expression  whose  radicand  is  integral. 

468.  If  the  radicand  is  a  fraction  : 

1.  Multiply  both  terms  of  the  fraction  by  the  smallest  number  that 
will  make  the  denominator  a  perfect  power  of  the  same  degree  as  the 
radical  index. 

2.  Factor  the  new  radicand  in  such  a  way  that  the  denominator  may 
be  removed  under  Case  I,  or  under  Principle  2,  §  463. 

Examples 

1.    Vi=V|=V^6  =  iV6. 


2.    ^^  =  V^-  =  -s/(-2f)-2=-%J/2. 

3  rs —       o        3 1 


s    /5  a      2x  7IO  ax2     2  x  7  1       1A     . 


2x        1      3/T?c — = 

3      2x2 

3x 

4.    Va_1  =  \-  =  etc.    Let  the  student  complete  the  solution. 
*a 

EXERCISE 

469.    Change  to  equivalent  radical  expressions  having  integral 
radicands : 

1     %J~I<L.  5'    VA'  9'   « Va"1  +  b-K 

\16x2 


2.    x.l5"3 


6  a5  „     3  .  /l3~a# 

3.   66VSJ. 

4.  ^i  8-  ~syI-% 


12. 


336  Radicals 

Change  to  equivalent  radical  expressions  having  integral  radir 
cands  : 


13. 


JK.  i8.    JIEL.  23.    J*±-C. 

i4.  JpL  19.  J^r+n?,       24  JEi, 


+  6 


i6;  //£.         9.1  ./i  oa  *c&? 


2/  ^(a-6)5 

21.    J^§L.  26.    Jj? 

17.    -gEJ.  22.    J^HZ.  27.    X/E^. 

V*-l  \     aft  A(6  +  c)3 

470.  Case  III.  To  reduce  a  radical  to  an  equivalent  radical 
with  a  smaller  radical  index. 

1.  ^-(o?)i=  0i  =  -</«. 

Let  the  student  state  the  definitions  and  laws  on  which  this 
reduction  is  based. 

2.  >^=</(a&2)«  =  \/a62. 

3.  V8a^3  __  V2a^.  Cancel  the  factor  3  from  the  radical 
index  and  from  the  exponents  of  the  factors  of  the  radicand. 
(Why?) 

It  is  clear  from  these  examples  that  this  reduction  depends 
upon  the  definitions  and  the  laws  of  exponents,  and  that  it  is 
possible  because  the  radical  index  and  the  exponents  of  all  the 
factors  of  the  radicand  have  a  common  factor. 

Thus,  in  example  2,  the  index,  6,  and  the  exponents  of  the  factors  of 
the  radicand,  2  and  4,  have  a  common  factor  2. 

471.  If  the  radical  index  and  the  exponents  of  all  the  factors  of  the 
radicand  have  a  common  factor,  that  factor  may  be  canceled  from  the 
index  and  from  all  the  exponents. 


Reduction  of  Radicals  337 

EXERCISE 

472.    Reduce  the  order  of  the  following  wherever  possible : 


1. 

Va2&2. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 

16. 

V9  a26<\ 

17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 

Va3  +  a4. 

2. 

-v/a2  +  b\ 

a/27  m6n3. 

aV  •  (a2)3. 

3. 

</{a  +  by. 
■^64  a3. 
a/64  a2. 

a/16  +  9. 

A^27  a3». 

4. 
5. 

Va2  +  a6. 
a/125. 

A^J    A/^. 

^32a« 
a/o^2. 

a/256. 
V625. 

6. 

a/12  afy». 
</IaJ>. 
•y/9  a266. 

^343. 

7. 

A/c-2n  .  C14n( 

8. 

V#n  •  xn+1. 

473.  Simplest  Form  of  a  Radical.     A  radical  is  in  its  simplest 
form  if : 

1.  The  radicand  is  integral. 

2.  The  radicand  is  as  small  as  possible  ;  that  is,  contains  no  factor 
that  can  be  removed  from  under  the  radical  sign. 

3.  The  radical  index  is  as  small  as  possible. 

474.  Corresponding  to  the   three   parts   of   §  473  radicals 
may  be  simplified  as  follows  : 

1.  If  the  radicand  is  a  fraction,  simplify  by  Case  II. 

2.  If  the  radicand  contains  a  factor  that  may  be  removed, 
simplify  by  Case  I. 

3.  If  the  radical  index  can  be  reduced,  simplify  by  Case  III. 

475.  Real  numbers  occur  in  five  different  radical  forms. 

1.  Radicals  that  are  rational  numbers,  Vl6  =  4. 

2.  Radicals  in  their  simplest  form,  \/7,  y/£. 

3.  Radicals  with  fractional  radicands,  \/jt=iTv/^* 

4.  Radicals  of  which  a  factor  of  the  radicand  can  be  removed  from 
under  the  radical  sign,  \/28  =  2V7. 

5.  Radicals  of  which  the  radical  index  may  be  reduced,  y/a2  =  Va. 


338  Radicals 

EXERCISE 

476.  In  the  following  list  of  radical  expressions,  each  of  the 
five  forms  given  in  §  475  is  included.  Examine  each,  telling  to 
which  class  it  belongs,  and  simplify  all  that  are  not  simple. 

16.  ^^648.  29.   ^ieR. 

17.  -y/ax2  —  bx2.                          /o^iY 
30.    \- 

18.  ab<  <*  +  *  ^  2n 


1. 

Va3. 

2. 

Va4. 

3. 

</a2. 

4. 

■VaW. 

5. 

Vo=i. 

6. 

(a4b)K 

7. 

-\/27azx. 

8. 

2V|f 

* 


a2b2 


31-    </ih- 


19.   V(a»-aV).  32<      H^y 


20. 


21.  |V5|.  33.  (yf^p. 

22.  V9^V°.  34.  '^/^ 
9.    (5)*#                    23.    VTxy^.  35.  -V5a5*+i. 

24.  Va4  +  66.  36.  Vx^K 

25.  V^T^-  37.  V55* 

V'50  63 


10. 

V(o  -  6)3. 

11. 

1    /ll    2 

12. 

Vi  +  f 

13. 

V3"1  -  4-1. 

14. 

V432. 

26.    20  ft8- 


38.    Va2mcOT. 


5 


39.    tyx2n~Yn' 


27.    -VI  -25-1.  4o.    V(«2-2/2)2w. 


lS»i 


J^  +  y  +  tf)^       41.     Vm» 
15.    -^a3(1^6p.  \  3  42.   xVa^. 

43.  Is  the  diagonal  of  a  square  whose  side  is  5  inches  a 
rational  or  an  irrational  number  ?  Express  the  length  of  the 
diagonal  in  its  simplest  radical  form. 

44.  Is  the  diagonal  of  a  rectangle  whose  sides  are  3  inches 
and  4  inches  rational  or  irrational  ? 

45.  Which  of  the  radicals  of  examples  1  to  42  are  rational 
numbers  ? 

46.  Is  V5  +  V4  a  surd  ? 


Reduction  of  Radicals  339 

477.  Case  IV.  To  change  a  radical  to  an  equivalent  radical  of 
a  higher  order. 

1.  -y/a  =  a*  =  a«  =  -\/a2.     Let  the  student  explain. 

2.  Vo63  =  (abrf  =  (a63)To  =  *J/(5&»)»=  VaW\ 

478.  From  these  examples  we  have  the  rule : 

To  change  a  radical  to  an  equivalent  radical  of  higher  order,  multiply 
the  radical  index  and  the  exponents  of  all  the  factors  of  the  radicand  by 
the  required  multiplier.    (Compare  with  Case  III.) 

Examples 
1.   Change  VS  to  a  6th  order  radical. 

The  radical  index  must  be  multiplied  by  3  to  make  a  6th  order  radical. 
V3  =  #P=#27. 


2.   Change  3  V5  az3  to  equivalent  radicals  of  the  4th  and  the 
6th  orders. 

3V5owc3  =  3y/2baW  =  3\/l25a3z9. 

EXERCISE 

479.    Change  the  following  radicals  to  equivalent  radicals  as 
indicated : 

1.  Change  V5  to  4th  order  ;  to  6th  order. 

2.  Change  V3  ab2  to  6th  order ;  to  9th  order. 

Va2 
—  to  6th  order ;  to  8th  order. 
o 

4.  Change  V2  to  15th  order. 

5.  Change  -\fa2  to  8th  order ;  to  2d  order. 

6.  Change  5  to  a  radical  of  2d  order. 

7.  Change  V3  and  ^  to  radicals  of  6th  order. 

8.  Change  the  radicals  in  7  to  12th  order. 


340  Radicals 

CJiange  the  following  radicals  to  equivalent  radicals  as  indicated  : 

9.   Can  V3  and  a/5  both  be  changed  to  radicals  of  lower 
order  than  the  6th  ? 

10.  Change  Vll  and  v5  to  radicals  having  the  lowest  com- 
mon radical  index. 

Solution.     vTl  =  11^  =  11$  =  vT33l. 

#6  =  5*  =5*  =$'§5. 

11.  Eeduce  a/3  and  V5  to  radicals  having  the  lowest  com- 
mon radical  index. 

12.  Write  Va^,  V#22/2,  Va^/3  with  the  lowest  common  radi- 
cal index. 

13.  Eeduce  a/3,  a/5,  a/4  to  surds  with  the  lowest  common 
radical  index.     Also  2V2,  a/4,  a/8. 

14.  Reduce   to   lowest  common   radical  index  s/a—b  and 

15.  Reduce  to  radicals  of  the  same  order  2?,  3*,  5*. 

16.  Which  is  greater,  V2  or  S^/3  ? 


Solution. 

V2  = 

:#8. 

^3  = 

:^S. 

.  •.  \/3  is  greater. 

Which  is  greater  : 

Arrange  in  order  of  magni- 

17.   V5  or  aVII  ? 

tude  : 

18.    V5or  ^/12? 

22.    V2,  V6,  a/2|. 

19.    V|- or  -v7!? 

23.    V5,  aVIO,  VI5. 

20.    a/25  or  Vll  ? 

24.    V3,  </4,  aVS. 

21.    Q)t  or  (f)*? 

25.    VS,  a/10,  a/20. 

26.   How  do  we  compare  the  values  of  radicals  of  different 
order  ? 


Addition  and  Subtraction  of  Radicals        341 

480.  Case  V.     To  introduce  a  coefficient  under  the  radical  sign. 

a V6  =  VaWb  =  V^6.     (§  463.) 

To  introduce  a  coefficient  under  the  radical  sign,  raise  it  to  a  power 
corresponding  to  the  radical  index  and  multiply  it  by  the  radicand. 

Example 

EXERCISE 

481.  Introduce  the  coefficients  under  the  radical  sign: 


1.  3Va.  3.   2a2Vor6.  5.   5aVa-b. 

2.  a^/2.  4.  i  •  2*.  6.    (a  -  b)  Va+6. 

9.   Which  is  greater,  2V3  or  3V2? 

10.  Which  is  greater,  2  V2  or  3^3? 

11.  How  does  the  transformation  of  radicals  in  Case  V 
compare  with  the  transformation  in  Case  I  ? 

ADDITION  AND  SUBTRACTION  OF  RADICALS 

482.  Similar  Radicals.  Kadicals  which,  when  reduced  to 
their  simplest  form,  have  the  same  index  and  the  same  radi- 
cand are  similar  radicals. 

\/3a,  2V3a,  and  (a  +  b)VSa  are  similar  radicals.  Also  V3,  V27, 
V75  are  similar  radicals,  since  V27  =  3V3  and  V75  =  5V3. 

Show  that  V18,  Vi,  and  (50  a2)  2  are  similar  radicals. 

483.  Similar  radicals  are  added  and  subtracted  with  refer- 
ence to  their  common  radical  part  as  the  unit  of  addition. 

1.  2V3  +  5V3  =  7V3. 


342  Radicals 

2.  2Vl2+V3a2-  §V3  =  4V3  +  aV3-fV3 

=  (¥  +  «)V3. 

3.  3V2  +  Vl8-V24+V| 

=  3 V2  +  3V2  -  2 V6  +  iV6  =  6 V2  -  |V6. 

To  add  or  subtract  radicals : 

1.  Reduce  all  radicals  to  their  simplest  forms. 

2.  Add  or  subtract  the  coefficients  of  similar  radicals,  and  join  dis- 
similar radicals  with  their  respective  signs. 

EXERCISE 

484.    Simplify  and  combine  as  much  as  possible  the  following 
radical  expressions  : 

1.  V2  +  VT8  +  V50.  3.    V50+V72-V8. 

2.  V8  +  V32-V72.  4.    Vl28  -  V32  -  V18  -  V2. 

5.  V32+V50  +  V72+V25. 

6.  V63+V700-V175  +  V28. 

7.  V147-V192+V108-V125. 

8.  2V54-W96-3V75-^4. 

9.  -5V675  +  3V27+2VI2-4V432. 

10.  12*  +  75*  -  108*  +  48i       11.    a/16 +\/54+ a/2. 

12.  2\/l6  4-^250  +  4\/128-  2^54. 

13.  \/24  +  ^375  -  a/1029. 

14.  V2+^l6+V60+^2-aV2. 

15.  a/128 +V72-V50 -^54. 

16.  V80  + 3^1029 +  4^/81+ 2 V32. 

17.  4V8 -a/875  -3VI8  +  2aVI89. 

is.  vi26-yi+jy!. 

19.  Vi-Vi+VA-V27+V9. 

20.  ^V|+iV40-3V}-\/ip. 


Addition  and  Subtraction  of  Radicals        343 

22.  V^+VI+^i  +  T-s^^OO. 

23.  2Va3  +  3Va&2~. 


24.   3V63a63-VH2tt36a. 


25.  9V3^  +  24a;V3^§  +  lGV3^.  • 

26.  ^135 -3-^^40  +  5^/^^20-7^^625. 

27.  3  6sV^c  +  -VaV-c*x& 

28.  ^cri?-  s!/27a^  -  ^-  125  a2«. 

29.  Vx  +  3^Jx-2V^+2x?-($x)*+Vl2x. 

30.  2  a^  -  Va^a  +  V(a;  -  l)2a. 


31.  4V3a-7Vl2a2  +  5V48a+6V27a^-5V75a". 

32.  7- 24* -5- (-81)* +  5^1^+2^375: 


33.    5V16  +  5V-54-6V-128+7V-250  +  2V432. 

34.  2v/i+w--^+vt  +  vTv 

35.    2^-Vl8+iA/4-Vf|+iv/4. 


36.    V(a  +  b)2x  +  V(a  -  b)2x  -  2  Va2x. 


37.    V4+4a;2  +  V9  +  9x2+Va2+a2a;2-5vT+a-2. 


;        /3  a;' +  30  a;2  +  75  a;        ji 


^  /3a;!  +  30a;2+  75a;      _  /3a?  -  6x*  +  3a; 


18  \  2 

39.  3  V125  m37i2  +  nV20ra3  -  V500  ra3n2  -  mV45  mn2. 

40.  Find  in  the  simplest  radical  form  the  sum  of  the  six 
diagonals  of  three  squares  whose  sides  are  respectively  1  inch, 
2  inches,  and  3  inches. 


344  Radicals 

MULTIPLICATION  AND  DIVISION  OF  RADICALS 

485.  To  multiply  or  divide  monomial  radical  expressions. 

Va .  Vb  =  Vab  and  Va  •  Vb  =  Vab.     (§  463.) 
Also  Va  -  Vb  =yj^  and  y/ct  +  Vb  =y[^ .     (§  463.) 

Examples 

1.  Vfa  •  VITa  =  VWtf  =  7  a V2. 

2.  3^/a26  .  \/a&*  =  3^W  =  3  a&\/&2. 

486.  To  multiply  or  divide  two  monomial  radicals  of  the  same  order : 

1.  Make  the  product,  or  the  quotient,  of  the  coefficients  of  the  given 
radicals  the  coefficient  of  the  result. 

2.  Make  the  order  of  the  result  the  same  as  that  of  the  given  radicals. 

3    Make  the  product,  or  the  quotient,  of  the  radicands  the  radicand  of 
the  result. 

4.   Simplify  the  resulting  radical  expression. 

Note.    If  the  radicals  are  not  of  the  same  order,  reduce  them  to 
equivalent  radicals  having  the  same  radical  index.     (§  478.) 

Examples 

1.  Multiply  SVa  by  -  2-\/5a?. 

Solution.      3Va  •  (-  2  J/5a?)  =  -Qtytf  ^260*     (Why?) 

=  -6v/26^     (Why?) 
=  -6av'25^. 

2.  Divide  3 Vx  by  -  2lJ/Ib*. 

Solution.    3Vx  -§.( -21^41?)=-  l\lz£zz     (Why?) 

7   *  16  0* 

=  -^v'4».     (Why?) 


Multiplication  and  Division  of  Radicals       345 


EXERCISE 


487.   Perform  the  indicated  operations,  giving  all  results  in  the 
simplest  form : 


1.  V3-V6. 

2.  V3-VI2. 

3.  v~2-n. 


4.  V14+V36. 

5.  V5a;-7-Vsc. 

6.  S^T"5-^50. 


7.  V3-5V8. 

8.  V2-V3-V6. 

9.  7^-42^.3^. 


10.  2*.  4*.     (4*=(4*)i) 

11.  (Vd3)2. 

12.  VZ&.-y/toL 

13.  -y/Vx-J/dtf. 


14.  </26^.  V-50y5. 

15.  V| -V}. 

is.  VS-Vfi 


21.  2V5^-5V2. 

22.  Va6  -r-  V&#. 

23.  Va2^ -s- 2  Vafc2. 

24.  2-V2.     (2=  VI.) 

25.  a-rVa, 


To"- 


'10 


17.  Vfi 

18.  V12-V6. 

19.  V54-T-V3. 

20.  5V7H-2V5. 


26. 


27. 


12  a       lb 
Wa       JlOa 


28.    Va  +  6  Va  —  b. 


29.    (Va^6)». 

30.    (V3.  V4-  V5)-=-(VlO.  V6). 


31.  VI -V2. 

32.  Vf-V6. 

33.  ^f-Vf. 

34.  V^  •  VJs. 

35.  Vf-Vf. 

36.  (V2a^)2. 

37.  V2.V3-VI. 

38.  \/J.^3. 


39.  V|-^6. 

40.  (2-v^5)2. 

41.  Va2  —  62  -f-  cVa  +  6. 

42.  Va&2  —  62c -h  Va  —  a 


43. 


44. 


ac 
63d3 


a^^l    /a& 

\hr\ 


We 


a2c* 


346  Radicals 

Perform  the   indicated  operations,  giving   all  results  in  the 
simplest  form : 


45.   .6+J2  +  2+VS  51.   sJf.lJ 


4a^ 

6   ' 


46.    VlO.  V15+V7-  V42.  _       g      3/8^ 


52. 


y/r 


47.  V2  •  VI  +  </2  •  Vi. 

48.  \/7  •  ^/^49  +  V49.  g3      ZIJ^  t  5  3/4j^ 

49.  (3V2)3W2.  ^252/5      ^5y' 


50.    1J**  •  ? JE.  54,     -** 

2\3  2/2     4^2^ 


V2a3 


488.  To  multiply  radical  expressions  when  one  or  both  factors 
are  polynomials. 

The  method  and  the  form  of  the  work  are  the  same  as  in 
the  multiplication  of  polynomials  in  Chapter  V. 

Multiply  each  term  of  the  multiplicand  by  each  term  of  the  multi- 
plier, simplify  all  results,  and  combine  the  terms  as  much  as  possible. 

Examples 

1.  (3V2  +  VI2  -  2  a/5)  V2  •=  6  +  2V6  -  2^/200. 

2.  Multiply  (3  +  V5  +  2V~6)  by  ( V5  +  V6). 

3  +  V5  +  2V6 

3V5  +  5  +  2V30 

12+   V30  +  3V6 
3V5  +  17  +  3V30  +  3V6 

3.   Expand  (V2+^3)2. 

In  this  exercise  multiply  by  type  form. 

( V2  +  V2f  =  (  V2)2  +  2  V2^3  +  (■&§)'. 
=  2  +  2^72+^/9. 


Multiplication  and  Division  of  Radicals      347 

EXERCISE 
489.    Multiply  the  following : 

1.  (2  V3  +  3  Vi  +  4  V6)  V3. 

2.  (4V5+2V6+V20)4V5. 

3.  (3Vl8-h4VT2+5V56+V27)2V3. 

4.  (4V25+3V75  +  2Vl8)2V3. 

5.  (V6+^2-2^/5)V3. 

9.   (3^4  +  4^  +  2^/6)^/10. 

10.  (2  Vl+\/J-  2VS  +  4v/27) V£- 

11.  ('V5a5x-V20aS+-^^N)\/5a». 

13.  (9  a2  -  6 Va3a  +  15^a^)3^a6a?. 

14.  ( Va2  —  a2  +  V(a  —  a)"1) Va  —  a;. 

15.  (7  +  2V6)(9-5V6). 

16.  (5VI4  +  3V5)(7V14-2V5). 

17.  (Vl2-2V7)(2H-V21). 

18.  (2V7-5V6)(^-2V6\ 

19.  (2V2-V8)(3V2  +  5V8). 


348  Radicals 

Multiply  the  following  : 

20.  (2  V5  +  3  V2  -  8  V6)  (2  +  5  V2  -  3  VIS). 

21.  (5V8  +  3V18  +  5V27)(9V6-12V18). 

22.  (</2  -</S)  (-^4  +  ^9). 

Apply  type  forms  to  multiply  examples  23  to  32,  and  to  others 
when  convenient. 

23.  (a*  -  &*)  (a*  +  &*). 

24.  (Vi5-Vii)(Vl5+ViI). 

25.  (Vl0+V6)2;   (V2-1)2. 

26.  (V5-V2)(V5+V2)-v/l6. 

27.  (V2-1)3. 

28.  ( Va  +  Va  —  b)  (  Va  —  Va  —  b). 

29.  (Va^+Va~+~&)2.  31.    (3V2  -  5)(3V2 -f  7). 

30.  (6V5-5V3)2.  32.    ( V2  + 1)  ( V2  +  3). 

33.  (V2-l)(2+V2  +  l). 

34.  V3-  V2(V24  +  iV96+V486). 

35.  V5-V2V5+V2.  37.    ^8-VlO:v'8  +  VlO. 

36.  V4+V7V4-V7.  38.   yj \/ . 

39.    (Va*-Va^  +  V^)(Va+V2/). 
40.    (V3-1)4.  41.    (-2V5-V3)*. 

42.  Find  the  value  of  x2  +  x  —  1  when  x  =  —  1  +  V5. 

43.  Find  the  value  of  2  a2  —  5  a;  +  2  when  x  =  — ±- — -  • 

4 

44.  Is  _  5  +  VlO  a  root  of  4a;2  +  20a;  +  15  =  0  ? 

45.  Find  a  mean  proportional  between  7  —  Vl3  and  7  -f  Vl3. 

46.  Find  a  fourth  proportional  to  3,  2  +  V5,  2  —  V5. 


Rationalizing  Factors  349 

47.  Find  a  third  proportional  to  2  and  V2  +  V-J-. 

48.  Solve  2  :  ( V5  -  1)  =  (|  V5  -  J)  :  x. 


Ha — T    /a  —  1  _,  .     /la        /l  .  a 

49.    Va2-1\^— — -•  51.    */-._-.*/-  +  _. 

a  +  1    /3  a3      a2 -4    /3  a 
*    a-2\2  6*  '  a2-1^263' 

54.    (aWx  -  a  VaJ3  -  J  V^)  (4  Va  -  -  Va3). 

CL 

55.    (m2Va5*3-mVa^)2.  56.    (aVy +  bVtf)2. 

57.    (3V^-2V^)2. 


a  +  3     /5m  #  4s2  —  1    /  64 t?6 
2a;_l\4ri2'    9-a2\l25m3 


5  +  2a    jm2  —  b2  t  4 m2  —  n2   jx2  —  y2 
2m- n*  y  +  x    '  4a2-  25^ b  +  m  ' 


.   yWa2  •  V^a. 


60 


RATIONALIZING  FACTORS 
ORAL  EXERCISE 
490.    Multiply  the  following  : 
1.  V3-V3.       2.  ^/4-V2.        3.  -s/a.J/tf.        4.  Va~3-Va. 
5.    (2--V5)(2+V5).  6.    (V5-V2)(V5+'V2). 

7.  (2V7+3V5)(2V7-3V5). 

8.  (aVb  +  xVy)  (aVb  -  xVy). 

9.  (V6-1)(V5  +  1). 

10.    (2V^-3Vy)(2-v^  +  3V^). 


350  Radicals 

491.  The  product  in  each  example  of  §  490  is  a  rational 
number. 

Rationalizing  Factor.  If  the  product  of  two  irrational  factors 
is  a  rational  expression,  either  of  the  factors  is  a  rationalizing 
factor  of  the  other. 

If  two  binomial  quadratic  surds  of  the  form  a^/b  +  x^/y 
differ  in  one  sign  only,  either  is  the  rationalizing  factor  of  the 
other,  for  (a  V&  -f  x  Vy)  (a  V&  —  x  Vy)  =  a26  —  xhj,  a  rational 
expression. 

492.  Two  binomial  quadratic  surds  that  differ  only  in  one 
of  their  signs  are  conjugate  quadratic  surds. 

Thus,  2  —  V3  and  2  +  V3  are  conjugate  quadratic  surds.  What  is 
their  product? 

DIVISION   OF   POLYNOMIAL  RADICAL   EXPRESSIONS 

493.  In  the  ordinary  division  of  a  polynomial  by  a  mono- 
mial we  have, 

(12a2  -  15adW  3  a  =?iP--  ^^  =  4a-  56. 

v  J.  3a        3a 

Similarly,    (2  VlO  -  3  V3)  -h  V2  =  ^®  -  ^ 

V2         V2 

=  2V5-fV6. 

494.  To  divide  a  polynomial  radical  expression  by  a  monomial,  di- 
vide each  term  of  the  polynomial  by  the  monomial  and  simplify  the  result. 

EXERCISE 

495.  Find  the  quotients  : 

1.  V15-2V3. 

2.  4V12-2V3. 

3.  (V6  +  4V18-8V2)W3. 

4.  (V72+V3-4)W8. 


Division  of  Polynomial  Radicals  351 

5.  (3Vl5-V20+Vl0-7)^-2V5. 

6.  (2V32  +  3V2  +  4)-4V8. 

7.  (V8+^12+</2)-2V2. 

8.  (6  +  2V3-Vl8)W6. 

9.  Solve2:3-V7  =  4+V28:<c. 

10.  (42V5-30V3)-=-2Vl6. 

11.  J1  +  -JI. 

\'a3     ^a 

12.  Va?x  ■+■  Va^. 


13.  (a  —  6)-=-Va&;  (a  —  6)-r-Va  —  6. 

14.  (l_Vo)-f-  Vl-Va. 

15.  (acy2  Vz2  —  ac?2;  Vy)  ■+■  a  vVz2. 

ORAL  EXERCISE 

496.    IFto  is  £&e  smallest  rationalizing  factor  of  each  of  the 

following  ? 


1. 

Va. 

2. 

Va. 

3. 

</5». 

4. 

Va». 

5. 

2*. 

6. 

8*. 

7. 

2*. 

8. 

A 

9. 
10. 

3  V2  ax2. 
Va-6. 

16. 

17. 
18. 

sJ2a2\ 

\    c2 

1-V2. 

11. 

Va3  -  a2b. 

3a—  V&. 

12. 

2V7. 

19. 

2V»  +  3Vy. 

13. 

2^21. 

20. 

Va  —  V&. 

14. 

v&. 

21. 

4  _  3  Vd*. 

15. 

.  n  Vm3. 

22. 

cVw  —  d  Vv. 

23.  According  to  what  type  form  of  multiplication  should 
two  conjugate  quadratic  surds  be  multiplied  ? 

24.  Can  you  give  more  than  one  rationalizing  factor  for  a 
given  radical  expression  ?     Try  with  Va36 ;  Vl2. 


352  Radicals 

RATIONALIZING   DENOMINATORS 

497.  Division  by  radicals  may  be  performed  by  a  method 
known  as  "Rationalizing  Denominators."  When  the  divisor 
is  a  binomial  or  a  polynomial,  this  is  the  only  practical  method. 

Study  the  following  examples  : 

1.  2  -j-  V 2  =  — -  =  — — =  -— —  =  V2.     Explain. 

V'2      V2-V2        2 

2.  (5-V3H(5+V3)  =  ^^  =  (5-V^(5-^) 

5+V3      (5  +  V3)(5-V3) 

^28  -  10V3  _  14  -  5V3 
22  11 

3.  3^-2^4  =  ^=    3^/2_  =  3jg 

2</4     2^4-^2         4 

498.  The  preceding  examples  illustrate  the  rule. 
To  divide  when  the  divisor  is  a  radical  expression  : 

1.  Indicate  the  division  in  fractional  form. 

2.  Multiply  the  numerator  and  the  denominator  by  the  smallest 
factor  that  will  produce  a  rational  denominator. 

3.  Reduce  the  resulting  fraction  to  its  lowest  terms. 

EXERCISE 

499.  Find  the  quotients: 

1.  3-V6.  7.  3-^2. 

2.  8h-2V2.  8.  2-s-^9. 

3.  V6-V3.  9.  5^12-4^/24. 

4.  3-2V3.  10.  8^-3^5. 

5.  9W3.  11.  (1+2V2)W3. 
.6.  5  h- 2^4.  12.  (2  +  3V3)W5. 


Rationalizing  Denominators  353 

13.  (3  +  4V3)-V6.  17.  4-(3+V3). 

14.  (6  +  3V6)W(L  18.  (3+V7)-(3-V7). 

15.  (8V3-2V5)h-3V2.  19.  (4-V8)-=-(V3+ v'5). 

16.  3-K2  +  V2).  20.  2V3^-(3-V3). 

21.  (3V7  +  4V6)-(2V^-3V6). 

22.  (V2  +  V3)h-(3V2  +  2V§). 

23.  (V2+V3+V5)-f-(V2-V3). 

24.  (^3-^2)-V3.  27.    (ah*  +  ah?)  +  ahk 

25.  (8  +  5Vl2)-=-3V72.  28.    l-s-(Vl2  -  V3). 

26.  (d*  +  6*)-l-(a*-&*).  29.    3-(V2+V3-2). 

Solution.     3(V2  +  V3  4-2) =  3(  V2  +  V3  +  2) 

(V2+V3-2)(V2+V3+2)  1  +  2V6 

_  3(V2  +V3  +  2)(1  -  2V6)       6-15V2-9V3-12V6, 
(1  +  2V6)(1-2V6)  ~23 

Two  rationalizing  factors  are  required  when  the  divisor,  in 
its  simplest  form,  is  a  trinomial  quadratic  surd  expression. 

30.  (2  +  V3)-(V2-V3+V5). 

31.  (1_V3)-(1+V2+V3). 

32.  5-V-V4+V3. 


Solution.     5-*-V4+V3  =  5         =  _5V4-V3        _ 

V4  +  V'S      V4  +  V3  •  V4  -  V3 
=  5  V4  -  V3  =  5  V13  V4  -  Vl  _  5  V52  -  13  V3 
Vl3  13  18 

33.  V3-V4+V3.  35.    12-V6-V10. 

34.  12-V6-VH.  36.    V5^VV6~  +  Vl. 


354  Radicals 

Find  the  quotients  : 

37.  2+(a-Va*-4). 

38.  (Va  +  b  +  Va  -  6)-h(Vo~4&  -  Va^6). 

39.  (4a2Vr+^)-(3aV(l  -  a)2). 

40.  (V2+Vl8+V50)-s-V2. 

Rationalize  denominators  in  the  following  fractions : 

«•  4'  «.  vg+vg. 

1  ^ 

42-    -ir==-  e„        V2+V3 


43. 


8V3-  2  V5  ,-        /r 
= •                            e,        Va  —  V6 


3V2 


51. 


cV#  —  d  Vy 

2  V2  +  3V3  9 

TTi 52-  " — 

*Vd  Va24&+Va2-& 

4V2+3V3+2V5  ^     V^^2_yq2T5-2 

2V3  Va2-62+Va2  +  62 

4^4  4-5^12 -hV§  kr    2Vl^P-3Vr=T2 

3  _  55.     • 

V2  Vl-62+Vl-C2 

L±LXl.  56     a;Vm3 — n2  +  ?/  Vm3  —  n2 

4  +  V5  #Vra3— ra2-  2/Vra3  — rc2 

500.    To  find  the  approximate  value  of  a  fraction  having  a  radical  in 
the  denominator. 

Divide  1  by  V2. 

This  might  be  done  in  either  of  the  following  ways : 


44. 


45. 


46. 


47. 


48. 


Involution  and  Evolution  of  Radicals        355 

_^  =  V2  =  L414^=707!.. 

V2       2  2 

The  second  method  is  much  to  be  preferred.     Why  ? 
When  approximate  values  of  such   quotients  are  desired, 
change  them  to  a  form  having  a  rational  denominator. 

EXERCISE 

501.  Find  values  of  the  following,  correct  to  two  decimal 
figures,   Mowing    that    V2  =  1.4142  ».f     V3  =  1.7321 .,    V5 

=  2.2361 .-.,  V6  =  2.4495 .... 

t     _3_  5.    (7-V5)-K3+V5). 

V3  6     V3+V2 

2.  2-h(V2-1).  '    V"3"V^ 

7.  V6+(V3-V2). 

3.  3V5-5-2V3. 

8.  — — -  when  a  =  2. 

4.  (3  -  V5)  -s-  (5  -  2  V5).  «+Va 

jPmcZ  £/ie  roote  correct  to  three  decimal  places : 
9.    2  a;  =  1  +  #V3.  10.    x V5  =  12  +  a;. 

11.   a;Va  —  a  =  a;V&  —  b,  when  a  =  2,  6  =  3. 

INVOLUTION  AND  EVOLUTION  OF  RADICALS 

502.  Powers  and  Roots  of  monomial  radical  expressions  can 
be  found  by  using  fractional  exponents  and  the  laws  of  ex- 
ponents. Powers  may  be  found  also  by  the  ordinary  multi- 
plication of  radicals,  as  in  §  486. 

1.    (2\/«2)2  =  (2  •  x$)2      (Definition  of  fractional  exponents.) 
=  4  •  x*     (Laws  4  and  3  of  Exponents.) 
=  4^a7     (Why?) 
=  4xy/x.     (Why?) 


356  Radicals 

=  [(8a;2)^     (Why?) 
=  (8  a2)*  (Why?) 

=  ^8^  (Why?) 

Note  that  in  taking  a  root  of  a  radical  expression,  a  coeffi- 
cient must  be  introduced  under  the  radical  sign,  but  it  is 
better  not  to  do  this  in  finding  a  power  of  such  an  expression. 

EXERCISE 

503.    Perform  the  indicated  operations : 

1.  (Vz)2;  (V^)3;  (V2)\ 

2.  (Va^a)2;  (Va-v^)3. 

3.  (2^/3)2;  (2V3)i 

4.  (2V3)4;  (2VS)*J  (2-^3)*. 

6.  (^a2)*;  (^a2)2.  12.    (^(a  -  y)2)4. 

7.  V^B";  V^I  13.    (-^(a2  -  b*)*)\ 


9.    (V3-V2)2. 


15.  (V (a2  -  2/2)2m)\ 

16.  (V^-V6)2. 

10.  (V5-V2)*.  17.    (^/^r7)6. 

11.  ( V(x  +  2/)3)4.  18.    (--^a2-62)9. 

SQUARE  ROOT  OF    A  BINOMIAL    QUADRATIC  SURD 

504.  It  is  sometimes  possible  to  express  the  square  root  of 
a  binomial  quadratic  surd  as  a  binomial  surd.  The  process  of 
finding  the  square  root  of  a  binomial  surd  is  readily  under- 
stood by  reversing  the  direct  process  of  squaring  a  binomial  surd. 


Square  Root  of  a  Binomial  Quadratic  Surd    357 

Thus,  ( V3  +  V2)2  =  3  +  2  V6  +  2  (1) 

=  5  +  2v6.  (2) 

This  may  be  compared  with 

(a  +  6)2  =  a2  +  2  ab  +  62.  (3) 

The  trinomials  in  (1)  and  (3)  are  both  perfect  trinomial  squares,  and 
the  trinomial  in  (1)  pears  the  same  relation  to  ( V3  +  V2)  that  the  tri- 
nomial in  (3)  bears  to  a  +  b.     From  (3)  we  readily  write 


Va2  +  2  ab  +  b2  =  a  +  b. 
and  from  (1)  we  may  write  in  the  same  way, 


V3  +  2  V6  +  2  =  VS  +  V2. 


If  we  are  asked  to  find  V 5  +  2  V6,  the  problem  evidently  reduces  to 
that  of  changing  5  +  2a/6  into  the  trinomial  3  +  2  V6  +  2.  This  may  be 
done  by  finding  two  factors  of  6  whose  sum  is  5.  These  factors  are  3 
and  2,  and  we  write 

V5  +  2V6"  =  V.S  +  2V6  +  2 
'  =V3+V2. 

505.   To  find  the  square  root  of  a  binomial  quadratic  surd : 

1.  Change  the  binomial  to  the  form  a  ±  2Vb. 

2.  Find  two  factors  of  b  whose  sum  is  a,  and  write  these  two  factors 
as  the  first  and  third  terms  of  a  trinomial  equal  to  the  original  binomial. 

3.  Extract  the  square  root  of  this  perfect  trinomial  square. 

Strictly  speaking  V5  +  2 V6  is  ±(V3+V2).  If  we  are 
concerned  with  the  positive  root  only,  it  will  always  be  ob- 
tained by  writing  the  larger  factor  of  b  as  the  first  term  of  the 
trinomial. 

Examples 

1.   Find     V7+V40. 

Solution.  V7  +  ViO  =  V7  +  2VlO  (Step  1  of  the  rule. ) 

=  V5  +  2\/5^2  4-  2    (Step  2  of  the  rule.) 
=  V5  +  V2.  (Step  3  of  the  rule. ) 


358  Radicals 

2.    Find     Vl9  -  4  Vl2. 


Solution.         Vl9  -  4V12  =  Vl9  -  2v^8 


=  V16-2Vl6.3  +  3 
=  4  -  V3. 

3.   Find     V6+VII. 


Solution.  V6  -f  Vll  =  V6  +  2  V-f 


=  W+V|  =  i(V22+V2). 

EXERCISE 

506.  Find  the  square  root  of: 

1.  3+2V2.  ii.  8+V39.    .  21.  12  ±  V44. 

2.  (6-2V5).  12.  9-V32.  22.  12  ±  V80. 

3.  7+2V6.  13.  9+V65.  23.  12  ±  V108. 

4.  7±V48.  14.  9  +  V80.  24.  12±Vl40. 

5.  8-V28.  15.  2+V3.  25.  10±Vl9. 

6.  9-V56.  16.  f+V2.  26.  10±V36. 

7.  7-V45.  17.  9  +  VI7.  27.  10  +  V51. 

8.  8-V60.  18.  9+V56.  28.  10±V64- 

9.  5  +  V9.  19.  9-V77.  29.  10  ±  V75. 
10.  8-V15.  20.  5+V2I.  30.  10  ±  V84. 

31.  Try  to  discover,  by  studying  the  form  of  these  binomial 
surds,  and  the  relation  of  the  numbers,  how  to  make  examples 
like  the  above. 

MISCELLANEOUS  EXERCISES  IN  EXPONENTS  AND  RADICALS 

507.  l.    (a)  (x  +  x-y.  (c)  (Va  +  Vy+Vz)2. 

(b)  (x  +  a>-i)(a>  -  x-1).  (d)  (a*  +  a?)(a?  -  a*). 

2.   4"^ +(£r2  + 21° +  9*,  3.   4  a0 +(4  a)0 +  4^0°. 


Review  of  Exponents  and  Radicals  359 

4.  Find  the  value  of  V#2  —  y2  when  x  =  a?  +  a~*  and  y  = 
ai  _  a~\ 

5.  Find  the  value  of  (x2  —  y2)0  when  x  =  3  and  ?/  =  1. 

6.  5V24-V54  +  3V96. 

7.  (a  -  6)  -s-  (a1  +  a*&*  +  &^). 

8.  Find  the  square  root  of : 

x-i  +  yi  +  2  a;~y  —  2x~hj  —  x~xy%.     (Yale.) 

9.  Simplify,  writing  the  result  with  rational  denominator : 

=— (Mass.  Institute  of  Technology.) 

#+Va2  +  a2 

10.  (a)  [(tf +  &fy*+(a* -**)*]■  J 

(6)  [(a*  +  &i)2+(a*-&*)2]2. 

11#    fl?"^+y~^  .  (a>Vty-     (Princeton.) 

Vai  +  Vy 

12.  2^  •  23 -f- 54~^.     (Yale.) 

13.  -s2(9-s2)^+V9^2+         3         .     (Yale.) 

3 


L      fx^ 


Vi- 


1  o 

14.   Simplify  x/^Y'.Y—')    •  (£\  .     (Princeton.) 

15-   3^  +  V40+J|--4=-     (Princeton.) 

16.  Find  to  3  decimal  places  — — 

. 4  +  V3 

17.  V17  +  12  V2. 

18.  .M±5_A§El+_2jLv?=7.     (Yale.) 
V*-0     Mz  +  j/     a?-i/2  *      v         y 


360  Radicals 

1Q     /    /64a266   .     3/     a^y      n. 

19-      \  ^ — r~»  +  \/ *    •     ^lve  answer  with  positive  ex- 

\  *  81  m6n2       \      m~V 

ponents. 

20.  Find    a    mean    proportional    between    V6  —  V2   and 

V6+V2. 

21.  Find  a  fourth  proportional  to  1,  2  +  V7,  2  —  Vf. 

22.  ^  ^i  —  4  a£  -f-  2  cc^  +  4  a;  —  4  x*  +  x%. 

23.  (a6"2c2)*(a362c-3)^  +  ^j-- 

24.  Simplify  (a)    Vl4  +  6^5. 

(b)  |Vl  -  x  +  a(l  -  a;)"*. 

^/2 2a/5 

(c)   .     Give  answer  in  simplest  radical  form. 

V3  +  V5 

(d)  SVl  +  SV^-Wi.     (Sheffield.) 

25.  (f)-i  -2-3  +(V6)i  +  128-^-(7V5)°. 

27.    Show  that  4""2 .  82""  .  2»  =  4. 

*.   Simplify  (^(%^)-*.     (Yale.) 

29.  Simplify  — — — ,  and  compute  the  value  correct  to 

V2-VI2 
two  decimal  places. 

30.  Simplify  (a)  £^£. 

(&)  3V|  +  2V3:-4V^.  (c)   ^2^W8^. 

(d)  2,^(1  +  4^-4^(1+4,)^      (Sheffield) 
(1  +  4*)* 


Review  of  Exponents  and  Radicals  361 

31.  Simplify  : 

(«)  (A)*  x  (iF)i  (c)  J^+Ji*-J™. 

*    X  "ft  *  o 

(b)  2(1  -  2  X)"*  +  3i/l-2x.     (d)  V\  -f-  VJ.     (Sheffield.) 

32.  Simplify  ^=H 1_ 


a-2 


1-V2*     1+V2a     l-2x 
33.    Simplify: 
(a)  V2-VI.  2V3-2a;-a;(3-2a;)-^, 

(6)   2^-^+<yl2  +  £±l.     (d)   ^ 

w      \a;      Vy      ^  a*/  v  ;    a~W 

_^3a-Vay 


9a2_[_1_6aya     /3a-Va\2 
x  x  Va; 


35.   3V28-V5V27+V60-|V112. 

36.  v|^i-v^-hT%+3vn. 

37.  V|ViVIV|  +  V480-Vi3j. 

38.  (V3+V2-1)*.     * 

39.  (V2-^2)3. 

15  +  6V5     7-2V5 
40     =~  ~ * 

•      2  +  V5         4-V5 

41.     (V^-2+^)H-(V^-^). 

20  +  30V2     5-2V2 


42. 


3+V3  2-V3 


2x-b     2b-x 

43.    Show  that        b^3    ,     ^3       =  V3. 
1      2a;  —  &      2ft  —  # 

&V3        a>V3 


XX.    RADICAL  EQUATIONS 

508.  A  radical  equation,  or  an  irrational  equation,  is  an  equa- 
tion in  which  the  unknown  number  is  involved  in  a  radicand. 

Thus,  3Vx  =  5,  Vic  +  3  =  5  are  radical  equations,  but  xV3  =  5  is 
not  a  radical  equation.     (Why  ?) 

Is  a?*  =  10  a  radical  equation  ? 

To  solve  a  radical  equation  it  is  generally  necessary  to  ra- 
tionalize the  equation.  The  following  examples  will  illus- 
trate the  method : 


1. 

V3»-5  =  0. 

Solution. 

V3x  =  5.     (Why.) 
3 x  =  25.     (Squaring  both  members.) 
x  =  8f 

Check. 

VS  •  8J  -  5  =  V25  -5  =  0. 

2. 

14+V2^=16. 

Solution. 

V2x  =  2. 
2x  =  4. 

x  =  2. 

Check  mentally. 

V. 

3. 

£  +  40  —  3  =  7  —  Vx. 

Solution. 

Vx  +  40  =  10  -  Vx.     (Why?) 

ar  +  40  =  100-20Vx  +  x. 

20Vz  =  60. 

Vx  =  3. 

x  =  9. 

Check  mentally. 

Radical  Equations  363 


4.    (a)  aj-4-V»-hl6  =  0.        (b)  x  —  4  +  -y/x  +-  16  =  0. 


Solution.  x  —  4  =  Vac  + 16.  3  —  4  =  —  Vx  +  16. 

32-83  +  16  =  x  +  16.  x2-8x  +  16  =  3  +  16. 

x2  -  9  x  =  0.  32  -  9  x  =  0. 

a(a;  -  9)  =0.  3(3  -  9)  =0. 

3  =  0  or  9.  3  =  0  or  9. 

Check.  When  3  =  0.  Check.     When     3  =  0. 


0_4-V0  +  16=-4-4  =  -8.  0-4  +  V0  +  16=-4  +  4  =  0. 

.  •.  0  is  not  a  root.  .-.  0  is  a  root. 

Check.     When        3=9.  Check.     When    3  =  9. 


9-4-V9+16  =  9-4-5  =  0.  9-4  +  V9  +  16  =  5  +  5  =  10. 

.  \  9  is  a  root.  .  •.  9  is  not  a  root. 

If  the  principal  square  root  of  the  radical  is  taken,  3  =  9  satisfies  (a) 
but  not  (b).    Also,  3  =  0  satisfies  (6)  but  not  (a). 

509.  Extraneous  Root.     From  example  4,  (a)  and  (b)  it  is 

seen  that  in  solving  a  radical  equation,  roots  are  sometimes 
found  that  do  not  satisfy  the  equation.  Such  roots  are  extrane- 
ous roots. 

The  student  will  notice  that  in  step  2,  under  both  (a)  and  (6) 
the  equations  are  the  same  since  the  squares  of  V#  +-16  and 
—  Vx  +-16  are  the  same,  x  +- 16.  It  is  here  that  the  extrane- 
ous root  is  introduced. 

510.  To  solve  a  radical  equation  : 

1.  Arrange  the  terms  of  the  equation  so  that  a  radical  is  alone  in  one 
member. 

2.  Raise  both  members  of  the  equation  to  a  power  corresponding  to 
the  order  of  the  radical. 

3.  Solve  the  resulting  linear  or  quadratic  equation  by  the  usual 
methods. 

When  more  than  one  radical  occurs  in  the  equation  it  may 
be  necessary  to  repeat  steps  1  and  2  of  the  rule  one  or  more 
times. 


364  Radical  Equations 

Example 

Solve:    Vx  +  60  =  2 V#  +  5  +V<c. 

Solution.  x  +  60  =  4x  +  20  +  4Vx2  +T»  + •'«, 

40  —  4  x  =  4Vx2  +  5x. 
10  -  x  =  Vx2  +  5  x. 
100  -  20  x  +  x2  =  x2  -f  5  x. 
-  25  x  =  -  100. 
x  =  4. 
Check  mentally. 

EXERCISE 

511.    Solve  the  following  radical  equations: 

1.  Vx  =  3.  5.    \^s4 

2.  (2a^  =  4.  6.   a*  =  4. 

3.  ^/2^=4.  7.    Vz  =  a  +  Vb. 

4.  2+Va  =  5.  8,    V23  w  +  52  -  16  =  19. 


9.   8  V4  a  +  5  =  7  V7  x  -  13. 


10.  V4a  +  17  +  14  =  15  +  2V^. 

11.  V49  x  +  85  -  12  =  -  11  +  7V^. 

12.  Vp  +  9  +  H  =  104-Vp. 

13.  VaT+45  =  9-Va. 


14.    V(2a>-l)(2a>  +  3)=2a-l. 

15.  V32  +  a;  =  4+V».  19.   2V#  —  V2a;  =  2. 

16.  5V5-7  =  3V«-1.  20.    2V^  +  l  =  2V^  +  3 


17.    Vx  +  ^/2x  =  l. 


SVx  -  2      3 Va;  -  5 

21     V^4-29  =  Va  +  37 
18.    V^  +  V3«  =  2.  VaJ  +  5       V»  +  7 

22.    V^  +  4  =  V^  +  8_ 
V#  +  2      Va  +  5 


Radical  Equations  365 

23.  (18  -  VlO  -  V3(^  -  3))*  =  2. 

24.  (2V«  +  3)(2V^-3)=2. 


25. 
26. 

VI  +  16  x  4-  2  V14  +  4  a;  =  11. 

5x-9        V5^-3  .   - 
= f-  5. 

V5  a;  -  3             2 

27. 

V7a>4-2-   5x  +  6   • 
V7  *  +  2 

o 

28 

V14      a?  +  Vll      a?- 

Vll-a; 

29. 

^  +  12  a:2  _  x  +  4. 

30. 

Va;2  +  2  a;  -  14  =  Va;2  -  5  -  1. 

31. 

V5a;-4=V2a:  +  l  +  l. 

32.  V(a;  +  2)(aj-5)=2. 

33.  2^^0=V37 


34.    Va;  — 6+Va;- l=Va;— 9  4-Va7+6. 

Solve,  and  determine  whether  any  of  the  roots  of  equations 
35  to  43  are  extraneous : 


35.    V10  +  a+V10-a;=6. 


36.    VlO  +  x  -  VlO  -  x  =  6. 


37.  Vl2  a; +  109  =  2  a; +  3. 

38.  V3a>-5+VaT+"6  =  0. 


39.    Va;  -f  5  =  x  —  1. 


40.  3aj-4V»-7=2(a;-f  2). 

41.  Vl  +  a;  +  a;2+Vl  —  x  +  x2  =  V6. 

42-  \/3  +  \/^^  +  Va  =  2. 

43.  VaT+~3+V2a;-3  =  6. 


366  Radical  Equations 

Solve : 


V3x2  +  4+V2x2  +  l     7 
Hint.     Use  composition  and  division. 


45.  a;  +  4a;V4a;  +  5=(4a;+  l)V4a  +  5  —  2. 

46.  V3a?2-1+V3^^  =  a 
V3a2-1-V3^¥2      & 

47.  Find  a  number  which  added  to  its  square  root  gives  56. 

48.    g + g =  «. 

a  +  V2-a2     a;-V2-x2 

49.  5  n-1    aal       V5n-1 

V5^  +  l  "      2 


50. 


51. 


VJ  —  z  Vl  +  Z 

2-VI+2     2+Vl^ 

1  1  Jlx 


52.    V2  8  -  3  =  8  -  3. 


53.  V2«-5  +  6  =  ic4-2. 

54.  V2  a;  +  1  +  2  VaJ 


21 


V2oj  +  l 


t  _         Vm      .  3  —  Vm      5 

55.      —  -\ — — =  — 

3— Vm         Vm         • 


56.  -v3a2-6a  +  f  ==V5a-2a;2--2ir2-. 

57.  V(a-l)(3a>-6)=a-2. 


58. 


V2p 


Vp2  -  9      Vp  +  11 


59.    VIOT^  V*-2+1 
VlO  -  a?     V10  -  a; 


XXL    IMAGINARY  NUMBERS 

512.  Consider  the  equation  x2  +  4  =  0,  or  x2  =  —  4. 

This  equation  asks  the  question :  What  is  the  number 
whose  square  is  —  4  ?  There  is  no  rational  or  irrational 
number  that  will  answer  this  question,  for  all  real  numbers 
are  positive  or  negative,  and  their  squares  are  positive  num- 
bers. Hence  the  square  root  of  a  negative  number  has  no 
meaning.  A  similar  difficulty  arises  if  we  attempt  to  solve  the 
quadratic  x2  +  2  x  4-  2  =  0. 

In  order,  then,  to  make  the  solution  of  the  quadratic  equation 
general,  that  is,  always  possible,  we  require  a  different  number 
from  any  we  have  previously  studied. 

If  we  solve  the  equation  x2  -f  4  =  0,  or  x2  =  —  4,  by  the 
method  of  §  410  we  get  x  =  ±  V—  4. 

In  order  that  the  result  should  represent  the  solution  of  the 
equation,  ±  V  —  4  must  be  such  a  number  that  (  ±  V  —  4)2  =  —  4. 

We  define  ±  V—  a  as  such  a  number  that 
(±V^)2=-a. 

513.  Imaginary  Number.  An  even  root  of  a  negative  num- 
ber is  an  imaginary  number. 

In  the  present  chapter  we  shall  deal  only  with  the  square 
root  of  the  negative  number. 

514.  The  student  must  not  fall  into  the  error  of  thinking 
that  V— aV— a=V(— a)(— a)=  a,  as  would  be  the  case  in 
the  multiplication  of  ordinary  radicals.  We  are  now  dealing 
with  a  new  kind  of  number  which  does  not  always  obey  the 
laws  of  radicals  and  which  is  defined  as  such  a  number  that 
(  ±  V  —  tt)2  =  —  a.     This  number  is  wholly  different  from  an 

367 


368  Imaginary  Numbers 

ordinary  square  root.  Thus,  V9  =  ±  3,  and  V7  =  ±  2.645  — 
correct  to  three  decimal  places,  but  it  is  not  possible  to  find 
exactly  or  approximately  in  real  numbers  the  value  of  V  — 4. 

We  shall  always  deal  with  the  imaginary  number  as  the 
product  of  two  factors,  one  real  and  the  other  the  imag- 
inary unit,  V—  1. 

For  example,  V  —  4  =  2  V  —  1,  V— a  =  VaV—  1. 

To  further  facilitate  the  work,  we  introduce  the  symbol  i  for 
the  imaginary  unit  and  write  V  —  4  =  2  i  and  V  —  a  =  i  Va. 

515.  Powers  of  the  Imaginary  Unit. 

By  definition, .(  V^)2  =  —  1  or  i2  =  —  1. 
From  this  we  get  the  following : 

i=i,  fi  =  i}  &  =  ?  i»  =  ? 

i2  =  -l,  i6=-l,  i10  =  ?  ;14  =  ? 

*  =  -  j,  ^7  =  -  i,  in  =  ?  i15  =  ? 

*«1,  t*=sl,  l12=?  l16  =  ? 

Any  power  of  the  imaginary  unit  may  therefore  be  reduced 
to  one  of  the  four  numbers,  i,  —  1,  —  i,  1. 
What  is  the  value  of  i4  +  *6  ?  i7  -f  i"  ? 

516.  Complex  Number.  The  sum  of  a  real  number  and  an 
imaginary  number  is  a  complex  number. 

Thus,  2  +  i,  and  3  +  2  i  are  complex  numbers. 

a  -f  fo*  is  the  general  form  of  the  complex  number,  hi  is  the 
general  form  of  the  pure  imaginary.  In  either  of  these  a  and  b 
may  have  any  real  values. 

517.  Operations  with  Imaginary  Numbers. 

All  operations  with  imaginary  numbers  can  be  performed  by  first 
writing  the  numbers  in  the  general  form  and  then  proceeding  as  with  real 
numbers,  using  the  symbol  i  as  we  should  use  any  other  letter.  In  case 
higher  powers  of  /  occur  at  any  time  in  the  course  of  the  work,  they 
should  be  reduced  as  indicated  in  §  515. 

Po  not  leave  1  in  any  denominator;  that  is,  rationalize  the  denominator. 


Imaginary  Numbers  369 

Examples 
518.    1.    Reduction  to  General  Form. 

(a)  V^6  =  V-  1  •  b  =  V^IVb  =  i  VS. 
(6)  3-V:r2  =  3-iV2. 
(c)  2+V^4  =  2  +  2l 

2.   Addition  and  Subtraction  of  Imaginary  Numbers. 


(a)  V-9+V^16  +  V-25  =  3*  +  4t  +  5t  =  12i. 

(6)  V^2  +  \/^18  +  V^200  - -v^64  =  i  V2  +  3 i V2 

+  10  iV2  +  3^2  =  3^2  +  14  i  V2. 
(C)   V^44+V^99-V^176+V^275  = 

2i  vii  +  3i  vii  -  4^  vn + 5i  vn = 6i  vn. 

3.  Multiplication  of  Imaginary  Numbers, 
(a)  1*  =  *;  i10  =  -l. 

(6)  (1  +  t)(l  -  0  =  1  -  *  =  1  - (-  1)  =  2.     (Explain.) 

(c)  Vr=:12V^3  =  iVi2.iV3  =  i2V36  =  6t2  =  -6. 

(Explain.) 

(d)  V^3  .  V27  =  iV3  •  V27  =  tVST  =  9i. 

4.  Division  of  Imaginary  Numbers. 

(a)  V'=n*^V^2  =  2*V2--iV2  =  2. 
V75        5V3       5       5i  5i 


W 


2V^3     2tV5      2»     2*  2 


(c)         v^ [       =      ^  V2(2-tV2) 

2  +  V^2     2  +  * V2      (2  +  t  V2)  (2  -  t  V2) 

==2V^-2i  =  2V2  -2f=  V2- 
4-2*2  6  3 


370 


Imaginary  Numbers 


EXERCISE 

519.    Simplify  the  following  expressions,  according  to  the  gen- 
eral rule  given  in  §  517,  and  the  illustrative  examples  of  §  518. 

1.   Write  in  general  form : 

(a)  3V^3.  (d)  V3+V^3. 


(b)  7-V-5.                                 (e)  5+V-a2. 

(c)  3+V-4.                                 (/)  V-16. 

2.    V-9  +  V-  16-V-36+V-81. 

3.  V-16+V-4-V-9+V-144. 

4.  V:=^  +  V^r9  +  3?;4-Vl6. 

5.  V-2+V-72-V-32. 

6.  V-63+V-700-^-64. 

7.  V- 45 -3V20  +  4V- 80 +->!/- 125. 

8.  V^84  +  V^"f 

o 

13. 
14. 
15. 

9.    3&V-a3c  +  -V-  abc*. 
c 

10.  i  -f-  V  +  $  +  i*. 

11.  (V-2)»;  (V-3)4;  (-V-4).2 

12.  (V-l)100;  (V-l)101;  (V-l)102. 

V^-V-32.                       17.    (-V-a)V-6. 
V^3  •  V^4.                          18.    VHf '  V^5  •  V25. 

V^-V-5.                          19.    V-12-V3. 

16. 

(_V-5)V-125).               20.    V-25-V4. 

21.  (_V-3)(V-12)(V^4). 

22.  (_V^|)(-V"^2).(V^3). 

23.  (V::^3+V7  +  3V:r5)2V^3. 

24.  (_3V:^5  +  4V8-3V'=r7)(-4V^3). 

25.  (_l-}-V^3)2. 

26.  (-1+V^3)3. 


Imaginary  Numbers  371 

27.  (l+O3;  (l  +  O4. 

28.  (4  +  3 V:r2)(4  -  3V^2). 

29.  (Vl2  +  2 V^8)( V12  -  2V^8). 

30.  (V^+V^)-KV^2-V=3). 

31.  3V^8+(-2V^2);  6 V^^ -r- 4 V^~3. 

32.  5V28^-3V^7;  — ^=- 

-V-16 

33.  42-(3-2V^3). 

34.  (l  +  *)-(l-i). 

35.  (1+V=2+V3)  +  (1  +  V=3). 

36.  42-r-(2  2'V3  +  3tV6). 


37.   Find  the  value  of  x2  +  x  +  1  (a)  when  x 
(b)  when  cc  =  — — 


-l+V-3 


-3 


38.   Find  the  value  of  x  +  -  when  x 

x 


l  +  i. 


39.  i  +  i2  +  i3---  i8. 

40.  t  *  i*  •  i3  •••  i8. 

41.  (x*-cHy. 

42.  (ra3  +  atf)2. 

47.    (V5  + 


48.  (Vn2-V-w2)2. 

49.  (a*  —  bi)z. 

50.  (a  +  fo')3-(a 
2 


43.  (4-V^i)2. 

44.  (T+V^)2. 

45.  (_4  +  2fo')2. 

46.  (3  +  5  0(4-7i). 

"6)(V6-V^8). 

1-2  i  V3 


6i)5 


51 


52. 


3+V^2 

2/ 
3-f-2iV:r6 


53. 


54. 


55. 


56. 


1  +  2  tV3 


i-£ 
l 


i-i 

l 


+ 


l  +  i 
l  +  i 


l  +  i      1  —  * 


372  Imaginary  Numbers 

Simplify  the  following  expressions : 

57     3  +  2i  +  S-2i  7-24* 

3-2i     3  +  2»"  '     4-3i* 

x  —  iVy  _E     5  —  29iV5 

bo.     —  . 

7-3  i  V5 
1+33  i  V3 
4  +  3 1 V  3  ' 
V3  + 1 V2 

V3  -  ?V2* 

68.     /H,   1   .N    + 


66. 
67. 


4  +  3*  ' 
56  +  33i 

12-5*' 
1- 20  iV5 

69. 


63. ^.  70. 


(l  +  f)2      (i_f-)i 

1  1 

(i  +  o4    (i  -  ¥ 

■y/x  —  y  +  Vy  —  a; 


7—2  t  V5  Vx  —  y  —  Vy  —  a; 

71.    Va?  +  V-y      Vy+V^# 
Va?  —  V  —  2/      V.y  ~"  V  —  » 

#oZve  £/*e  following  equations,  writing  imaginary  answers  in  the 
general  form : 

72.  a2 +  2  #  +  2  =  0. 

Solution.  x2  +  2x=—2. 

Z2+2Z+1:=-1 

a;  +  1  =  ±  i. 

x  =  —  1  ±  i. 
Check.     (-  1  ±  £)2  +  2(-  1  ±  *)  +  2  =  +  2  i  -  2  ±  2  i  +  2  =  0. 

73.  #2+4oj  +  6  =  0.  78.  x2  +  x  +  1  =  0. 

74.  #2-4#  +  8  =  0.  79.  3#2  +  4  =  2#. 

75.  #2-2aa?  +  4a2  =  0.  80.  #2  +  5  =  4a\ 

76.  a;2 -4  #  +  7  =  0.  81.  #2  +  2#  +  4  =  0. 

77.  2#2  +  5a?  +  4  =  0.  82.  3a;2  -  10a;  +  10  =  0. 


XXIL    QUADRATIC  EQUATIONS 

{Continued  from  Chapter  XVI) 

520.  Equations  of  the  forms  x2  =  k,  and  ax2  +  bx  =  0. 

1.  What  is  a  quadratic  equation  ?     (§  405.) 

2.  What  is  an  incomplete  quadratic  ?     (§  407.) 

3.  What  are  the  two  forms  of  incomplete  quadratics  ? 

521.  In  the  solution  of  the  following  examples,  irrational 
answers  may  be  left  in  the  simplest  radical  form,  and  imagi- 
nary answers  in  the  general  form. 

EXERCISE 

1.  Give  the  rule  for  solving  the   quadratic  in  which   the 
first  degree  term  is  missing.     (§  410.) 

Solve  the  following : 

2.  #2  =  10.24.  5.  2a<aH-3)-s=5(»+l)-3. 

3.  (s  +  i)(a-i)  =  0.  6.  (x  -  S)(x  +  3)  =  1. 

4.  (*-3)(*+2)  =  19-*       7.  3_l  =  5  +  (#-5). 
v         /v         y  x     x  8 

8.  Give  the  rule  for  solving  the  quadratic  in  the  form  ax2 
+  bx  =  0.     (§  414.) 

9.  6x2  =  lSx.      • 

10.  (3a  +  l)2+(3a;-l)=0. 

11.  3<c2-ar\/8  =  0. 

12.  a?^3+3x  =  0. 

13.  x2  +9  =  0. 

14.  (V5-s)(V5  +  a>)  =  0. 

15.  0+V6)(a;-V6)  =  -2. 

373 


16. 

5       4       7 
2  #2     3~4a2' 

17. 

1 4-  a     #  +  25_0 
1  —  #     a;  —  25 

18. 

a+x     x+ b      * 
a—x     x—b 

19. 

ax2  =  a2(a +4  b)  +  4  a&2. 

20. 

4a;2  +  a;V— 1  =  0. 

374  Quadratic  Equations 

Solve  the  following : 

21.  *=-!.  24.    aa2-frP  +  c=£t 

a;  mx2  —  nx+p     p 

22.  (x  + 4)2  =  23.  a;.  ^    ^+2__3x-J  =  () 

23.  a2(b2-x)  =  b2(a-x)2.  3  a;  +  4       x  -  2 

26.  (2  x  +  7)(5  a-  9)  +  (2  »-  7)(5  a;  +  9)  =  1874. 

27.  (1  +  x)(2  +  a>)(3  +*)  +  (!-  a>)(2  -  x)(S  -x)  =  120. 
—  x       1  —  bx 


28. 


1  —  ax       b  —  x 


29.   *=  + 1 _*S. 

30.  V#  +  4  —  V5  a;  —  24  = 


Va;  +  4 


31.  2V5  +  2x-Vl3-6a;  =  y37-6a;. 

32.  If  a  quadratic  equation  lacks  the  absolute  term,  one 
root  is  zero.     Why  ? 

33.  The  roots  of  a  quadratic  in  the  form  x2  =  k  are  equal 
in  absolute  value  but  of  opposite  sign.     Why  ? 

COMPLETE  QUADRATICS 
522.   Solution  by  Completing  the  Square. 

What  is  a  complete  quadratic  ?     (§  408.) 

We  may  solve  complete  quadratics  by  three  different 
methods  ;  namely,  by  completing  the  square,  by  formula,  and  by 
factoring. 

1.  Which  of  these  methods  have  already  been  treated  in 
Chapter  XVI? 

2.  What  is  meant  by  the  p-form  of  the  quadratic  ?     (§  420.) 

3.  How  is  an  equation  reduced  to  the  p-i orm  ? 

4.  Change  ax2  +  bx  4-  c  =  0  to  the  ^p-form. 

5.  What  must  be  added  to  complete  the  square  ? 
(a)tf-3s+(    );  (b)x2  +  x  +  (    ) ;  (c)  ±x*  +  5x+  (    ). 


Complete  Quadratics  375 

523.  To  solve  a  complete  quadratic  equation  : 

1.  Reduce  the  equation  to  the  />-form. 

2.  Complete  the  square  of  the  first  member  by  adding  to  both  mem- 
bers the  square  of  one  half  the  coefficient  of  x. 

3.  Extract  the  square  root  of  each  member,  using  both  roots  in  the 
second  member  of  the  equation,  and  solve  the  resulting  linear  equations. 

Unless  otherwise  suggested,  the  irrational  answers   may  be  left  in 
simplest  radical  form. 

Example 

x-\-l  =  2x  —  1      3 
x  —  1       x  +  1 

Solution.    L.  C.  D.  =  xl  —  1. 

x2  +  2  x  +  1  =  2  x2  -  3  x  +  1  -  3  x2  +  3.     (Why  ?) 
2x2  +  5x  =  3.       (Why?) 
x2  +  $  a  =  f .       (The  p-form.) 

*+f*+H=f|.   (Why?) 

,-.  x=±|-f  =|or-3. 
EXERCISE 

524.  Solve  the  following  by  completing  the  square: 

1.  2x2  —  5x  =  S. 

2.  z2- fa +  1  =  0. 

3.  9«*  =  a?+f. 

4.  5^2=39+2y. 

+  i=l?  =  2.5. 


6. 

=  x  —  1. 

z  +  l 

7. 

z-1       1 
x  — 13      a; 

8. 

(*-5)(*-3)+a*-15=sG. 

9-z 


9.  /f?  +  5V9aj-l)  =  (3  +  5ar).4 

10.  (x  —  p)(4  x  —  5  p)  =  x2  —  p2. 

11.  g*C2»-fi)  +  _2_,3. 

2z-l         l-2z 

12.   a2-6a;  +  4  =  0.  13.   x2-2ax  +  b  =  0. 


376  Quadratic  Equations 

Solve  the  following  by  completing  the  square : 

14.  ax2  -2bx  =  c.  18.  2  a;2  + 15.9  =  13.6  x. 

15.  2x*  -f  5  x  +  4  =  0.  19.   a2  +  2a  =  0. 

16.  a2  +  a;  +  l  =  0.  20.   2  a;2  -  .21  a?  +  .001  =  0. 

17.  Verify  16. 

21    5^-l  +  3^-l  =  2  +  !K_1< 
9  5  a; 

22.      7-.8 4a?"5  =  2. 

ll_2a;     l-3a; 

5  +  a     8-3»_   2a; 
2o. 


3  —  x  x  x  —  2 

6  x  +  4     15-2a;_7(a;-l) 
5  a?- 3  5 

(Multiply  through  by  5  and  transpose.) 
M     »  +  l  ,     12    _a>-4  ,  17 
25'    "9~"l"aqT4"~4~+"6"' 

2 
a;  +  2  3  16  a; 


29. 
30. 


5ar(2a?-l)      x      -4a;24-l 

a; +  2       7a;-2      6a;24-9a;  +  5^0 
3a>-2     3a>  +  2  4-9a;2 

a?  +  a  |  a;  4-  &  .  a;2  —  ab  _  q 
a  b  ab 


x  -\-  a  —  b  ,  a;—  a  — 2  b  ,  a  4-  5 

31.      —  |  r  T 

a  o  x 

3«  _  3  a; -20  =  2      3  a;2  -  80 
2       18 -2a;  2(a;  -  1)  ' 

33.    V2a;4- 1  -r  8  «*  $• 


34.    V10  a;- 34  4-2  Va;  4-4=  V2(3  x  4-  35). 


Complete  Quadratics  377 


35.  j°  +  Vf=3- 

36.  Va  +  2  +  V2  a;2  H-  a;  =  2. 
Ill 


37. 


x  -  1     a?  -  3     35 


38.    ^ lm     4 


10  -  a;  x-7 

39.    (a?-2)(3oj  +  l)=10+(2aj  +  l)(aj-3). 

40.  Sirl+S-i.-i^Sf^l 

9  x  5 

In  examples  41  to  44  ,/md  £/ie  roots  correct  to  two  decimal  places : 

41.  a*  +  2a;-2  =  0.  43.   a;2  +  Sx-  11  =  0. 

42.  2L+!  +  ?L±i  =  0.  44.   aj  +  10+-  =  0. 
#—  3      a?-f-  2  x 

525.   Solution  by  Factoring.     The  equation  (x  —  a)(x  —  b)  =  0 
is  satisfied  when  x  =  a  or  x  =  6  and  not  otherwise. 

For  suppose  a;  =  a, 

We  then  have  (a  —  a)(a—b)  =  0, 

or  0  •  (a  -  6)  =  0. 

Therefore  the  equation  is  satisfied  when  x  =  a. 
When  x  =  bf  we  have 

(&  _  a)  (b  -  6)  =  0  or  (b  -  a)  0  =  0 
Therefore  the  equation  is  satisfied  when  x—b. 

To  prove  that  the  equation  is  not  satisfied  when  x  has  any- 
other  value : 

Suppose  x=  c,  where  c  is  different  in  value  from  both  a  and  h 

.-.  (c-a)(c-6)=0. 

We  should  now  have  the  product  of  two  factors,  neither  one 
of  which  is  0,  equal  to  0,  and  this  would  be  absurd. 


378  Quadratic  Equations 

526.  Section  525  furnishes  us  the  basis  of  the  factoring 
method  of  solving  equations. 

To  solve  an  equation  by  factoring  : 

1.  Write  the  equation  in  order  of  powers  of  the  unknown  number, 
and  with  the  second  member  zero. 

2.  Factor  the  first  member  into  linear  factors  if  possible. 

3.  Put  each  factor  equal  to  zero,  and  solve  the  resulting  equations. 

527.  This  method  of  solving  equations  applies  equally  well  to 
equations  of  higher  degree  than  the  second  if  the  factoring  can 
be  accomplished.  Also,  it  is  not  necessary  to  factor  into  linear 
factors,  for  factors  of  the  second  degree  with  respect  to  the  un- 
known may  be  put  equal  to  zero  and  solved  by  the  preceding 
methods  of  solving  quadratics. 

Example 

Solve  (3  a  +  2)(2 x  +  3)=(a> -  3)(2* -  4) 

Solution.     6  x2  +  13  x  +  6  =  2  x2  -  10  x  +  12.     (Why  ?) 
4z2+23x-6:=0.     (Why?) 
(4z-l)(z  +  6)=0.     (Why?) 
4s-i=0andx  +  6  =  0.     (Why  ?) 
x  =  |  or  —  6. 

EXERCISE 

528.  Solve  the  first  30  examples  of  the  following  set  orally. 
The  student  should  review  Cases  VI  and  VII  of  Factoring. 

Solve  : 

1.  (a;-4)(»-5)=0.  9.   a;2 +  13  a; +  40  =  0. 

2.  (a?  +  4)(a?-7)=0.  10.   x2  +  19a?  +  90  =  0. 

3.  (2a?-5)(a:-3)=0.  11.   x*  -  21  x  +  20  =  0. 

4.  (a;  +  |)(7aj-l)=0.  12.    a*  -  7a;  +  12  =  0. 

5.  x2  -13a +  30  =  0.  13.   a2- 7a +  6  =  0. 

6.  a2 +  13 a +  30  =  0.  14.    x2  +  16a  +  48  =  0. 

7.  a2 -13 a -30  =  0.  15.   a2- a -42  =  0. 

8.  a2  +  13a -30  =  0.  16.   a2  +  a  -  42  =  0. 


Complete  Quadratics  379 

17.  x2-2x  =  63.  27.   3a2-2a-l  =  0. 

18.  a2 +  2  #  =  35.  28.   4z2- 4#  +  1  =  0. 

19.  a2 -JO  a  =  39.  29.    (x  -  1.25) (a  +  .75)=  0. 

20.  a2-10a;  =  0.  30.   a2- f  a +  i  =  0. 

21.  x*-  16  =  0.  n.   52±I  +  -li-.-«+L 

22.  ax2  —  a  =  0. 

23.  (ar-a)(a;-a)=0.  32.    -^  - -A_     ,  2. 

24.  a2-6a  +  9  =  0. 

25.  a  (a -5)  =36. 


9  2#+3 

a?  —  3      a  — 2 

26.    *-7  =  *2-  34         1      =J2*~3 

*  *  -  1       V  » - 1 

35.    Vl  +  2/  +  Vl  -  V  =  V3  -  y. 


33 


36.    V4  —  a? -h  Vl  +  «  =  V9  +  a;. 
_2a-3  =  0  40    3r-2 

(x-1)2       x-1  r  r  +  1 


37.    -i ^-3  =  a  4Q>    3r-2_2r-i  =  2> 


38  ^2_?_^  +  /2  =  0  41-    x2  +  3ax  =  abx  +  3a?b. 

P         9        92 
1  -  o  42.    (a2-9)(a;  +  3)=0. 

39  4-      -1      =  2 

a;-3     x-  2     2'  43.   3aa  +  12a2Va;  =  15a3 

a_l      ^-2      1 

44. =  -  • 

x  +  1  a  3 

45.  (x-a)(«+«)=4-a2- 

46.  (x  —  a)(x  —  b)=ab  —  a—b  +  l. 

47.  (x  -  l)2  =  a  (a2  -  1). 

48.  a2(6  -  a)2  -  62  (a  -  x)2  =  0. 

49.  a2  -  a2  -  (a  -  x)  (b  +  c  -  x)  =  0. 

50.  a2(a-a)2  =  62(5  _  ^  52.   gp+l-a+l. 

x  a 

51.  a*-(a  +  &)*  +  a&=:0.  53.     aa;2  +  bx  +  c  =  *■ 

a-^x2  +  ^a;  -|-  cx     Cj 

54.    (a-7)(2aj  +  5)-(3a;-l)(a:-7)=0. 


380  Quadratic  Equations 

Solve  : 


55.    U--\(2x-3)=4,x-l. 


x 


56.    (7-x)x=ll--\(3x  +  8). 
Kft     3x     3a-20      „  .  3a2-8 

57. =  ZH • 

2       18-2a>  2a-2 

529.  General  Form.  The  general  form  of  the  quadratic  equa- 
tion is  ax2  +  bx  +  c  =  0,  where  a,  b,  and  c  are  any  numbers. 

1.  3x2  —  7  a?  +  5  =  0  is  in  the  general  form.  In  this  equa- 
tion a  =  3,  b  =  —  7,  c  =  5. 

2.  (a2-f-&2)a;2+(a-Z>)a;  +  3  =  0.  Here  a  =  a2  +  62,  6  =  a-  b, 
and  c  =  3. 

3.  Change  to  the  general  form 

(ax  —  b)(c  —  d)  —  (a  —  6)  (esc  —  d)  x. 
acx  —  adx  —  be  +  bd=  acx2  —  bcx2  —  adx  +  bdx.     (Expanding.) 
—  acx2  +  bcx2  +  acx  —  bdx  —  be  +  bd  =  0.     (Transposing. ) 
(6c  -  ac)  x2  +  (ac  -bd)x-  (be  -  bd)  =  0.     (Collecting.) 
.•.  a  =  be  —  ac,  b  =(ac  —  bd),  c  =  —  (be  —  bd) . 

530.  Any  quadratic  equation  may  be  changed  to  the  general 
form  ax2  +  bx  +  c  =  0.  The  steps  are,  in  general,  clearing  of 
fractions,  expanding,  transposing,  and  collecting  terms. 

EXERCISE 

531.  Reduce  each  of  the  following  equations  to  the  general 
form  ax2  -f  bx  -f  c  =0  and  determine  the  values  of  a,  b,  and  c,  in 
each  case : 

1.  9a;2 -f  4a  =  325.  7.  ax2-bx=c. 

2.  17  a;2  =  418.  8.  ax2- (a2  +  l)x  +  a  =  0. 

3.  x2-2aa;  +  &  =  0.  9.  (a  -  x)2  =  0. 

4.  x2-ax  =  0.  10.  (x  +  a  +  b)2=0. 

5.  a2  —  a;  =  2  +  V2.  11.  abx2  —  a2x  —  b2x  =  ab. 

6.  x2  +  l=%x.  12.  Va;  +  16  =  x  -  3. 


Complete  Quadratics  381 

532.  Formula  for  Solving  Quadratic  Equations.  We  have  seen 
that  every  quadratic  equation  can  be  reduced  to  the  form 
ax2  +  bx  +  c  =  0.  The  solution  of  this  equation  leads  to  a 
formula  that  can  be  used  for  solving  any  quadratic  equation. 

ax2  +  bx  +  c=0. 

ax2  +  bx  =  —  c.     (Why  ?) 

a2  +  -a;  =  --.     (Why?) 

a  a  . 

*+£„,  +  £=£_£     (Why?) 
a        4  a2     4  a2     a 

"4ac      (Why?) 


4  a- 


^2a  2a 


_-6±V62-4ac 

37  — — • 

2a 

The  last  result  gives  the  two  roots  of  the  equation  whatever 
the  values  of  a,  6,  and  c  may  be.  The  roots  of  any  quadratic 
equation  can  therefore  be  found  by  substituting  the  values  of 
a,  b,  and  c  in  any  particular  equation  in  the  formula  for  the 
roots  ;  that  is,  in 


—  b  ±  V&2  —  4  ac 
2a 

This  formula  should  be  carefully  committed  to  memory. 

533.   To  solve  a  quadratic  equation  by  the  formula  : 

1.  Change  the  equation  into  the  form  ax2  +  bx  +  c  =  0. 

2.  Determine  the  values  of  a,  b,  and  c  for  the  given  equation. 

3.  Substitute  the  values  of  a,  b.  and  c  in  the  formula  and  simplify 
the  results. 


382 


Quadratic  Equations 


l.  Solve 

Solution. 


Examples 

6  a;2  +  x  =  15. 

6x2  ±x-  15  =  0. 


(General  form.) 


a  =  6,  b  =  1,  c  =—  15. 

g  =  -l±Vl-4.6.(-15) 


2-6 

12  2  3 

Check.  6  (f)2  +  f  =  V  +  f  =  15. 

2.    Solve     mx2  —  ra2a;  —  a;  +  m  =  0. 

Solution,      mx*  -(m2  +  1)*  +  m  =0. 

a  =  m,  b  =  —  (m2  +  1),  c  =  m. 


(m2  +  1)  ±  Vw4  +  2  m2  +  1  -  4  m2 
2m 
_(m2  +  l)j:(w2  -1) 


2m 


=  m  or  — 
m 


3.    Solve  a*-  7a -30  =  0. 


Solution.    »  =  L±^»±i20  =  7±18=10op-8b 

2  2 


EXERCISE 
534.   /Sofae  6y  £/ie  formula  : 

1.  a;2  -6»-  7  =  0. 

2.  a2  +8a?  =  20. 

3.  a2 -6a; -16  =  0. 

4.  a;(aj  +  5)  =  84. 

5.  a;2 +  7  a;  =  30. 

6.  a;2 -13a; +  42=0. 

7.  x2  —  19a  =  0. 

8.  x2  +  4  x  =  1. 

9.  2a2  4-  5a;  +  4  =  0. 


10.  9a;2  +  5  =  12a;. 

11.  36a;2  +  6  x  -5  =  0. 

12.  x2  +  8  a;  +  16  =  0. 

13.  4a;2- 4a; +  1  =  0. 

14.  a;2 -4  =  0. 

15.  a;2  +  |  a;  =  40. 

16.  3a;2  +  17a;  +  70  =  0. 

17.  a2  +  TV^  =  TV 

18.  20a;2 -2a;  =6. 


Complete  Quadratics  383 

19.  yi2-ic2-2«  +  12  =  0.  23.  ax2  +  2bx  +  c=0. 

20.  2  x2  —  f  x  =  1  24.  cm;2  4-  &x  =  c. 

21.  (2x4-l)2  =  0.  25.  x2  -  ax  =  0. 

22.  x2  +  ax  =  b.  26.  (x  —  l)2  —  ax2  4-  a  =  0. 

27.  (x-6)(x-5)+(x-7)(x-4)  =  10. 

28.  (2x-5)2-(x-6)2=80. 

29.  2x  +  ^  =  3.  31.    ^4-  — =  3. 

x  4       x 

30.  5-21=^=1.  32.    *"8         X"1 


4       4-x  x  +  2      2(x  +  5) 

33.  (a  —  x)(x  —  b)  +  a&  =  0. 

34.  (a  -  x)2  +{x  -  b)2  =  a2  +  &2. 

35.  x2  -  xV3  +  1=0. 

36.  Verify  35. 

37.  *=!-**=*  +  *=*  =  0. 

x  x—  1       x  —  9 

38.  x(a  +  b  —  x)  =  c(a  4-  b  —  c). 

39.  (n  —  p)x2  +(p  —  m)x  +  (m  —  n)  =  0. 

..     ax2  —  bx  -f  c 

40.    4—  =  c- 

x2  —  x  4- 1 

41.  (a  —  x)2  —  (a  -  x)(x  —  b)  —  (x  -  6)2  =  (a  —  6)2. 

42.  (a  -  x)3  +  (x  -  &)3  =  (a  -  6)3. 

Tri  ^e  following  equations  find  the  roots  correct  to  two  decimal 
places : 

43.   3x2-2x  =  40.      44.   x2-x-l=0.     45.    x2-30  =  -. 

3 

46.  Given   S  =  ±gt2  +  v0t.  When   S  =  200,   g  =  32,   and 
v0  =  10,  find  t. 

47.  x2  +  1.92  x- 3.83  =  0.  49.   3x2-4x-10  =  0. 

48.  x2  4- 3.14  x  4- 2.45  =  0.  50.  5(x2-  1)4- 10  x  =  7x2  -  15. 


384  Quadratic  Equations 

THEORY  OF  QUADRATIC  EQUATIONS 

535.   Relations  between  the  Roots  and  the  Coefficients  of  a 
Quadratic  Equation. 

Consider  the  equation 

ax2  +  bx  +  c  =  0. 
If  we  let  rx  and  r2  represent  the  roots,  we  may  write, 
_  —  b  +  V&2  —  4  ac 
Tl~  2a  ' 

2a 


—  5+Vfr2  —  4ac  .   —  6  — V&2  — 4ac         6 

.-.  rx  +  r2  = 1 =  —  - , 

2a  2a  a 


andrir2  =  -6  +  V&2-4ao.-6-V^-4ac  =  c. 
2a  2a  a 

The  student  should  work  out  these  results  in  detail. 

Therefore  we  have  : 

The  sum- of  the  roots  of  a  quadratic  equation  in  the  form  ax2  +  bx  -f  c 

b  c 

—  0  is  — ,  and  the  product  is  -  • 
a  a 

Examples 

1.  3a;2-7a;  +  2  =  0. 

The  sum  of  the  roots  is  |  and  the  product  is  f . 

2.  3a2 -3a:  =  7. 

3  x2  -  3  x  -  7  =  0.     .  \  n  +  r2  =  1,.  and  nr2  =  -f 
EXERCISE 

536.   Find,  without  solving,  the  sum  and  the  product  of  the  roots 
of  the  following  : 

1.  a;2 -21  a; +  20  =  0.  4.    4a;2  -  8a;  -  3  =  0. 

2.  a;2 +  7 a: +  12  =  0.  5.   x2-\x  =  \. 

3.  a;2 +  16  a; +  48  =  0.  6.  3a;2  -x  =  24. 


Theory  of  Quadratic  Equations  385 

7.  lx2  +  x  =  50.  12.    (x-3)(x-5)=0. 

8.  2x2  -14  x  +  23  =  0.  13.    3z2  +  ll  =  5a\ 

9.  5z2  +  13  =  14a>.  14.   5x2  =  12. 

10.  ax2  —  2bx  =  c.  15.   3  a;2  =  5  a;. 

11.  6a2  +  7a;  =  3.  16.   x2  +  px  =  q. 

17.  Show  that,  if  an  equation  is  in  the  form  x2  +  mx  +  n  =  0, 
r1-\-r2  =  —  m,  and  rx  •  r2  =  n. 

18.  What  is  the  sum  of  the  roots  of  an  incomplete  quadratic 
equation  of  the  form  x2  =  k  ? 

19.  One  root  of  x2  -f-  4  x  —  45  =  0  is  5.     Determine  the  other 
in  two  ways,  without  using  any  of  the  usual  methods  of  solving. 

20.  One  root  of  2  x2  —  7  x  —  15  =  0  is  5.     Find  the  other,  in 
two  ways,  as  in  example  19. 

537.   Nature  of  the  Roots  of  ax2  +  bx  +  c  =  0.     The  roots  of 
the  equation  a#2  +  bx  +  c  =  0  have  been  found  in  §  532  to  be 


_&  +  y&2- 

-  4ac 

2a 

_6_V62- 

-  4ac 

and  r2= 

2  a* 

Whether  the  roots  ^  and  r2  are  real  or  imaginary  (§  456),  and 
if  real  whether  they  are  rational  or  irrational  (§  457),  depends 
upon  the  expression  V&2  —  4  ac.     (Why  ?) 

1.  If   62  —  4  ac  =  0,  the   roots   are   real,  rational,  and   equal. 

(Why?) 

2.  If  b2  —  4  ac  is  a  negative  number,  the  roots  are  imaginary. 

(Why?) 

3.  If  b2  —  4  ac  is  positive,  the  roots  are  real  and  unequal,  and 
they  are  rational  or  irrational  according  as  b2  —  4  ac  is  or  is  not  a 
perfect  square. 


386  Quadratic  Equations 

538.  Discriminant.  The  expression  b2  —  4  ac  is  the  dis- 
criminant, since,  by  means  of  it,  we  determine  the  nature  of 
the  roots. 

1.  Determine  the  nature  of  the  roots  of  x2  +  5  x  —  6  =  0. 
Solution.     The  discriminant  is  25  —  4  •  1  •  (—  6)  =  49. 

.-.  The  roots  are  real,  rational,  unequal. 
Let  the  student  determine  the  roots. 

2.  Determine  the  nature  of  the  roots  of  9  x2  +  5  =  12  x. 
Solution.     9x2  —  12x  +  5  =  0. 

The  discriminant  is  —  36.     (Why  ?) 
.-.  The  roots  are  imaginary. 

3.  For  what  value  of  k  are  the  roots  of  x2  —  6x  +  k  =  0 
equal  to  each  other  ? 

Solution.     The  discriminant  is  36  —  4  k. 

If  k  has  a  value  that  satisfies  the  equation  36  —  4  k  =  0,  the  roots  will 
be  equal.     This  gives  k  =  9. 

Therefore  k  =  9  is  the  value  required  to  make  the  roots  equal  to  each 
other. 

If  k  <  9,  36  —  4  k  is  a  positive  number,  and  therefore  the  roots  will  be 
real. 

If  k  >  9,  36  —  4  k  is  negative  and  the  roots  will  be  imaginary. 

Let  the  student  determine  the  roots  when  k  —  9 ;  when  k  =  10  ;  when 
k  =  S. 

EXERCISE 

539.  Determine,  without  actually  solving,  the  nature  of  the 
roots  of  the  following  equations  : 

1.  a2 -7a +  10  =  0.  8.  3z2  +  2z  +  5  =  0. 

2.  12a2-x-l  =  0.  9.  z2-5a;  =  50. 

3.  3  x2  -  12  x  +  5  =  0.  10.  x2  -  5  x  +  50  =  0. 

4.  3z2-8;r  +  7  =  0.  11.  2a2-  7x  +  30  =  0. 

5.  2x2-5a-9  =  0.  12.  2z2-7z-30  =  0. 

6.  4.t2-13z  +  3  =  0.  13.  -7a2  +  22z  =  3. 

7.  25a2-  10 z  +  l  =  0.  14.  2x2  +  3  =  5a. 


Theory  of  Quadratic  Equations  387 

15.  x2—3x+k=0  when  k=2\.   17.  Answer  15  when  k  >  2\. 

1      5 

16.  Answer  15  when  k  <  2\.       18.  x  -j-  -  =  -  • 

X        £ 

19.  x  +  -  =  k,  when  k  lies  between  2  and  —  2  in  value. 

x 

20.  What  value  of  c  will  give  equal  roots  in2#24-4a;+3c=0? 

21.  Verify  20. 

22.  For  what  value  of  k  is  one  root  three  times  the  other  in 
X2  _  jcx  +.  75  =  3  ?  

Solution.     Here  n  =  3  r2,  or  *±^MZM  =  8  *  -  VF=800 
Solve  for  A;.  2  2 

23.  For  what  value  of  A:  does  one  root  of  x2  —  kx  +  40  =  0 
exceed  the  other  by  3  ? 

24.  Find  k  if  r,  =  7  r2  in  a2  —  foe  +  63  =  0. 

25.  Find  &  if  r,  =  2  r2  in  4  x2  -  9  a  +  k  =  0.     Verify. 

540.    To  form  an  Equation  with  Given  Roots. 

We  have  seen  (§  525)  that  (x  —  a)(x  —  b)  =  0  has  the  roots 
a  and  b  and  no  other  roots.  Similarly  (x  —  a)(x  —  b)(x  —  c)=  0 
has  the  roots  a,  6,  and  c  and  no  other  roots.  Thus,  we  can 
make  an  equation  with  any  required  roots. 

1.  Make  an  equation  whose  roots  are  2  and  3. 

Solution,  (x  —  2)(x  —  3)  =  0,  or  x'2  —  5  x  +  6  =  0,  is  the  required 
equation. 

2.  Form  an  equation  whose  roots  are  2  and  —  5. 
Solution,     (x  -  2)(x  +  5)  =  x2  +  3  x  —  10  =  0.     Explain   the  factor 

£  +  5. 

3.  Form  an  equation  whose  roots  are  f  and  -f. 

The  result  is  indicated  by  the  equation  (x  —  f  )(z  —  I)  =  0-    F°r  con- 
venience, multiply  by  15  in  the  form  of  the  two  factors,  3  •  5  ;  thus, 
3(x-!)5(z-f)=0. 
(3  x  -  2)(5  x  -  3)  =  0  or  15  x2  -  19 x  +  6  =  0. 


388  Quadratic  Equations 

4.  Form  an  equation  whose  roots  are  1  ±  V2. 
Solution,     (as  —  1  —  V2)(x  -  1  +  V2)  =  0,  or  x2  —  2  x—  1  =  0. 

5.  Form  an  equation  whose  roots  are  1,  —  1,  2. 
Solution,     (x  —  1)  (a  +  1)  (x  -  2)  =  0,  or  x3  -  2  x2  -  x  +  2  =  0. 

EXERCISE 

541.  Jfafte  equations  whose  roots  are  as  indicated: 

1.  2,3.  6.  -a,  -b.  11.  ±V3. 

2.  4,5.  7.  2,2.  12.  1±V5. 

3.  7,  -  1.  8.  ±  2.  13.  1  ±  i,  1. 

4.  0,6.  9.  a,  2  a.  14.  2,  1±2V3. 

5.  -  3,  -  2.  10.  6,  -  2  6.  15.  ±  i,  2. 

16.   2+V2,  3-V2. 

EQUATIONS   IN   THE   FORM   OF   QUADRATICS 

542.  Quadratic  Form.  An  equation  is  in  the  form  of  a  quad- 
ratic if  it  contains  two  powers  of  the  unknown,  one  of  which  is 
the  square  of  the  other. 

x*  —  5  x2  +  4  =  0,  x  +  x*  —  6  =  0,  and  x~$  +  x~^  —  6  =  0  are 
examples  of  quadratic  forms.  These  equations  may  be  solved 
for  a;2,  x^y  and  x*  respectively,  and  the  results  so  found  can 
then  be  solved  for  x. 

An  equation  may  be  in  quadratic  form  with  respect  to  some 
polynomial  containing  x. 

(x2  —  2)-f-  V#2  —  2  —  6  =  0  is  such  an  equation.  It  may  be 
solved  for  V#2  —  2  just  as  z2  -f-  z  —  6  =  0  may  be  solved  for  z. 

The  method  of  solving  such  equations  will  be  understood  by 
examples. 

1.   x4-  5  #2  +  4  =  0. 

(x2  —  4)  (x2  -  1 )  =  0.        (Solving  for  x2  by  factoring,) 
x2  =  4  or  1. 
x  =  ±  2  or  ±  1. 


Equations  in  the  Form  of  Quadratics         389 

2.   x  +  x*  —  6  =  0. 


x\  _  —  1  ±  v  1  +  24  _  _  3  or  2.     (Solving  by  the  formula  for  x^.) 
x  =  9  or  4. 

This  is  a  radical  equation;  4  is  a  root  and  9  is  an  extra- 
neous root. 

3.  aT*  +  afs  _  6  =  0. 

x~s  =  —  3  or  2.       (Solving  as  a  quadratic  for  x"1.) 
-I- =  -3  or  2.       (Why?) 

Xs 

xi  =  _  i  or  |.       (Why  ?) 

X=-^jOT^. 

This  is  a  radical  equation,  and  both  roots  satisfy  the  equa- 
tion. 

4.  (a?  -  2)  +  Va;2  -2-6  =  0. 
Here  we  regard  Vx2  —  2  as  the  unknown, 

^— -  _  _  i  ±  Vl  +  24  =  _  3  or  2.     (By  formula. ) 
2 
je2  -  2  =  9  or  4. 
z2  =  11  or  6. 
x  =  ±  y/TL  or  ±  V6. 

Do  the  roots  satisfy  the  equation? 

In  the  solution  of  such  an  equation  as  example  4,  it  is  some- 
times convenient  to  substitute  a  single  letter  for  the  expression 
we  are  regarding  as  the  unknown.  For  example,  we  might 
have  put  z  for  V#2  —  2  and  then  we  should  have 

z2  +  z  -  6  =  0. 

,  =  -l±V26  =  _8or8, 


Vx2-2  =  -  3  or  2,  etc. 


390  Quadratic  Equations 

a  —  x     x  —  b      a2  +  &2 


5. 


x  —  b     a  —  x         ab 


To .  a  —  x 


x  —  b 
Then  the  equation  takes  the  form 

*  +  l  =  «i+-L2. 

z        ab 
abz2-(a2  +  V2)z  +  ab  =  0. 
(az-b)(bz-a)=Q. 

whence  z  =  -  or  -. 
a       b 

^f  =  f.  Also  «  =  *=», 

x-b     b  x-b     a 

x=™?-.       (Explain.)  x  =  <*±*. 

a  +  b  *  a+b 

EXERCISE 

543.  Solve  the  folloiving  equations  as  quadratics,  substituting 
a  single  letter  for  an  expression  containing  the  unknown  ivhen 
desirable  : 

1.  x*  -13a;2  +  36  =  0.  8.    x4  -  21  a;2  =  100. 

2.  x\x*-  90)  +729  =  0.  9-    (a;2  -  10)  (a;2  -  3)=  78. 

3.  a;-7V*  +  12  =  0.  10-    (*2-5)2+(*2-l)2=40. 

11.   ar6-7ar3  =  8. 


4.   x  —  4  =  3a;i 


12.    9a*  =  9  + 2a;*. 


-     a;2      5      26 

5+5»     T"  13'   2^+Va^-3  =  0. 

6.    a*-9a?  +  8  =  0.  14'   2(Va;-3)2-3=V*. 

7     _2_       5=2  !5-   x^  +  Sx^  =  9x. 

a;2 +  3      x2       '  16.   2x^  -  3x%  +  a;  =  0. 

17.  (a;2  +  2 a;)2+  3(a;2  +  2  a?)  =  10. 

18.  fx  +  -\2-sfx+-\^-7  =  0. 

19.  ^±^+2^^?  +  3=0. 
a;2 -3        a;2 +  3 


Equations  in  the  Form  of  Quadratics 


391 


20.   60  —  Wx2  +  x  +  6  =  x2  +  a  +  6. 

Let  z  =  Vx2  +  x  +  6. 


21.   a2-2a;+6Vx2--2a;  +  5  =  ll. 
Hint.     This  equation  may  be  written 
O2 -2  x  +  5)  +  6>/a:-2  -  2  »  +  5  =  16. 


22.   x2  4-  5  =  8  a;  +  2  V#2  —  8  a;  +  40. 


23.  2.»2  +  3Va;2-a;  +  l  =  2a,'-t-3. 

24.  (2z2-3a;  +  l)2=22a;2-33a;+l. 

25. 


Ac-h*Y  +  4a;  +  -  =  12. 
\        xj  x 


26     *  +  s  +  5      *  +  s-5=ia 

tf  +  x-2      a2  +  a;  -  4 
Put  x2  +  a;  =  z. 

27.  2(a  +  3)(z  +  4)  =  (^  +  7z)(a2  +  7a;-3). 

28.  x2  =  8  Va2  +  16  -  32. 

29.  a?2 "+  V5  «  -f  a;2  =  42  —  5  ». 


30.    x2-9x-9Vx2-9x- 
3  3 


31. 


11  =-9. 

—  =4. 


x  +  V5  —  x2     a;  —  V5  —  a;2 


32.   2  a;10  =  3  a6 -a* 


33.  2a;2  +  3a;-5V2a;2  +  3a;  +  9  +  3  =  0. 

34.  (x2  -  2)*  -  4(a?  -  2)*  =  5. 

35.  9ar4+4ar2  =  5. 

36.  arJ-4Va^^2=-l. 

37.  V2~a7- 7x  =  — 52. 


38.   ar2-7a;  +  VarJ-7a;  +  18  =  24. 


392  Quadratic  Equations 

Solve  the  following  equations  as  quadratics: 


40.   ^H-lY+2^+i^  =  15. 


PROBLEMS  LEADING  TO  QUADRATIC  EQUATIONS 

544.    1.    The  sum  of  two  numbers  is  17  and  their  product  is 
42  ;  find  the  numbers. 

2.  The  sum  of  two  numbers  is  17  and  the  sum  of  their 
squares  is  185  ;  find  the  numbers. 

3.  Find  the  sides  of  a  rectangle,  knowing  that  its  perimeter 
is  52  feet  and  its  area  is  160  square  feet. 

4.  In  a  right-angled  triangle  the  measures  of  the  two  sides 
about  the  right  angle  and  the  hypotenuse  are  three  consecu- 
tive integers.     Find  them. 

5.  Same  as  problem  4,  if  the  sides  are  consecutive  even 
numbers. 

6.  The  sum  of  the  two  sides  about  the  right  angle  of  a 
right-angled  triangle  is  21  inches,  and  the  hypotenuse  is  16 
inches.     Find  the  sides  correct  to  two  decimal  places. 

7.  A  certain  rectangle  contains  216  square  feet.  If  both 
dimensions  are  increased  by  2,  the  area  is  increased  by  64 
square  feet.     Find  the  dimensions  of  the  rectangle. 

8.  The  sum  of  the  roots  of  a  quadratic  equation  is  3  and 
their  product  is  If.     Find  the  roots. 

9.  Same  as  problem  8,  if  the  sum  of  the  roots  is  2  and  their 
product  is  4. 

10.  Two  numbers  differ  by  2.1,  and  the  square  of  their  sum 
is  25.     Find  the  numbers. 

11.  Determine  the  positive  value  of  x  correct  to  two  decimal 
places  for  the  equation  x2  -f  y2  =  36,  knowing  that  5  y  =  27. 


Equations  in  the  Form  of  Quadratics 


393 


800' 

12.  Determine  the  positive  value  of  b  to  two  decimal  places, 
in  a2  +  b2  =  c2,  if  a  =  2.1  and  c  =  4.3. 

13.  Determine  the  larger  root  of  x2  =  .100  —  .200  x  correct 
to  three  significant  figures.     (Harvard.) 

14.  A  rectangular  box  is  3  inches  deep,  and  is  2  inches  longer 
than  it  is  wide.  Find  its  length  and  breadth,  if  its  volume  is 
105  cubic  inches. 

15.  A  rectangular  plot  of  land  is  600  feet  by  800  feet.  It 
is  divided  into  four  rectangular 
blocks  by  two  streets  of  equal 
width  running  through  it.  Find 
the  width  .  of  the  streets  if  to- 
gether they  cover  an  area  of 
67,500  square  feet. 

16.  What  is  the  width  of  the 
streets  in  problem  15  if  together  they  cover  one  fourth  the 
area  of  the  plot  ? 

17.  How  wide  a  strip  must  be  cut  around  the  outside  of  a  lawn 
60  feet  by  80  feet  so  that  the  strip  cut  may  contain  half  the  plot  ? 

18.  A  rectangular  tin  box  is  2  inches  deep,  and  is  2  inches 
longer  than  it  is  wide.  Find  the  length  and  breadth  if  it 
requires  88  square  inches  of  tin  to  make  the  box,  including 
the  cover,  making  no  allowance  for  waste. 

19.  If  from  a  certain  number  the  square  root  of  half  that 
number  is  subtracted,  the  result  is  25 ;  find  the  number. 
(Regents.) 

20.  The  numerator  of  a  fraction  exceeds  the  denominator  by 
2.  If  both  terms  of  the  fraction  are  increased  by  2,  the  value 
of  the  fraction  is  diminished  by  -^.     Find  the  fraction. 

21.  The  units'  digit  of  a  number  exceeds  the  tens'  digit  by  1. 
The  product  of  the  digits  equals  -§-  of  the  number.  What  is 
the  number  ? 

22.  In  an  automobile  race  of  462  miles  the  winning  car  runs 
2  miles  an  hour  faster  than  the  losing  car  and  wins  the  race  by 
•i-  hour.     What  is  the  winner's  rate  and  what  is  the  time  ? 


394  Quadratic  Equations 

23.  A  broker  buys  a  certain  number  of  shares  of  stock  for 
$960.  Later  the  price  falls  $20  a  share  and  he  finds  that 
he  might  have  bought  4  shares  more  for  the  same  money. 
How  many  shares  did  he  buy  ? 

24.  If  q  is  the  area  of  a  rectangle  and  p  is  the  perimeter, 

show   that  x2  —^-x  -\-q  —  0   is   an    equation    for  finding   the 
dimensions. 

25.  A  man  buys  apples  for  $  12.  If  the  price  had  been  20^ 
less  per  bushel,  he  could  have  bought  5  bushels  more  for  the 
same  money.  Find  the  number  of  bushels  bought  and  the 
price  per  bushel. 

26.  How  wide  a  strip  must  be  plowed  around  a  field  60  rods 
long  and  40  rods  wide  to  have  the  field  half  plowed  ? 

27.  A  stream  flows  at  the  rate  of  4  miles  an  hour.  A  man 
can  row  up  the  stream  10  miles  and  back  to  the  starting  point 
in  6  hours.  Find  the  rate  at  which  the  man  would  row  in  still 
water. 

Note.  It  is  to  be  assumed  in  this  problem  that  the  rate  at  which  the 
man  rows  upstream  is  equal  to  his  rate  in  still  water  minus  the  rate  of 
the  current,  and  that  in  going  downstream  his  rate  is  that  of  his  rowing 
in  still  water  plus  the  rate  of  the  stream. 

28.  A  motor  boat  goes  12  miles  up  a  river  and  returns  to 
the  starting  point  in  4|  hours.  Find  the  rate  of  the  current  if 
the  boat  can  run  7  miles  an  hour  in  still  water. 

29.  A  rectangle  is  6  inches  by  10  inches.  It  is  to  be 
doubled  in  area  by  equal  additions  to  the  length  and  the  width. 
Find  to  two  decimal  places  the  increase  in  the  dimensions. 

30.  Solve  29  if  the  area  is  to  be  made  four  times  as  great. 

31.  A  company  owns  two  factories  that  together  can  make 
252  automobiles  in  12  days.  Working  alone  one  factory  requires 
7  days  longer  than  the  other  to  make  this  number.  Find  the 
number  of  days  for  each  factory.     (Yale.) 


Review  of  Equations  395 

32.  If  the  product  of  three  consecutive  numbers  is  divided 
by  each  in  turn,  the  sum  of  the  quotients  is  191.  Find  the 
numbers. 

33.  A  man  having  bought  an  article,  sells  it  for  $  21.  He 
loses  as  many  per  cent  as  he  gave  in  dollars  for  the  article. 
How  much  did  it  cost  him  ?     (Yale.) 

34.  Find  the  price  of  eggs  when  if  two  less  were  given  for 
30  ^  the  price  would  be  2  ^  per  dozen  higher.     (Amherst.) 

35.  If  a  ball  is  thrown  vertically  upwards  with  a  velocity  v0, 
the  distance  in  feet  to  which  it  will  rise  in  t  seconds  is  given 
by  the  formula  d  =  v<$  —  i  gt2.  (g  =  32.)  Solve  this  equation 
for  t  when  v0  =  200,  and  d  =  300. 

REVIEW   OF   EQUATIONS 

545.  Equations  may  be  classified  as  to  their  degree  into 
three  groups : 

1.  Linear  Equations.     (Simple  or  first  degree  equations) 

2.  Quadratic  Equations.     (Second  degree  equations) 

3.  Higher  Degree  Equations. 

546.  Equations  may  be  classified  as  to  form  into  three  groups: 

1.  Rational  Integral  Equations. 

2.  Rational  Fractional  Equations. 

3.  Irrational  Equations. 

547.  In  solving  an  equation  we  begin  by  simplifying  as  much 
as  possible,  including  such  steps  as  expanding,  clearing  of 
fractions,  rationalizing,  transposing,  and  collecting  terms.  All 
these  steps  aim  toward  some  particular  form.  The  form  will 
depend  upon  the  kind  of  equation  we  are  solving  and,  in  the 
case  of  quadratic  equations,  the  method  we  intend  to  use  in  its 
solution. 


396  Quadratic  Equations 

Exceptions  to  the  general  directions  just  given  include  radical 
equations  solved  as  quadratic  forms  without  rationalizing,  and 
fractional  equations  solved  by  substitution. 

Skill  in  selecting  the  best  methods  of  solving  equations,  and 
in  discovering  methods  of  simplifying  the  work  will  be  gained 
by  experience  and  by  a  conscious  effort  on  the.  part  of  the 
student  to  achieve  such  ends. 

Higher  degree  equations,  if  they  can  be  solved  at  all,  by  the 
methods  of  elementary  algebra,  must  come  under  "  Quadratic 
Forms  "  or  under  the  factoring  method. 

REVIEW   QUESTIONS 

548.    1.  What  is  a  simple  equation?  a  quadratic  equation? 

2.  What  is  a  rational  integral  equation?  an  irrational 
equation  ? 

3.  State,  in  full,  the  three  different  methods  of  solving 
quadratics. 

4.  To  what  form  do  we  reduce  the  quadratic  when  we  solve 
by  "  completing  the  square  "  ?   by  formula  ?     by  factoring  ? 

5 .  Can  the  formula  be  used  to  solve  an  incomplete  quadratic  ? 

6.  What  do  we  mean  by  the  "  nature  of  the  roots  "  of  a 
quadratic  ? 

7.  How  do  we  determine  the  nature  of  the  roots  of  an 
equation  without  solving  the  equation  ?  the  sum  of  the  roots  ? 
the  product  ? 

8.  How  do  we  form  an  equation  with  given  roots  ? 

9.  What  do  we  mean  by  the  extraneous  roots  of  a  radical 
equation  ? 

10.  How  many  roots  has  a  quadratic  equation  ? 

11.  Knowing  that  3  and  —  5  are  the  roots  of  x2  -\-  2  x  —  15  =  0, 
can  you  at  once  write  the  factors  of  x2  +  2x— 15  ? 


Review  Exercise  397 

12.  Translate  into  verbal  language  the  condition  that 
ax2  +  bx  -f-  c  =  0  may  have  equal  roots ;  the  condition  that 
x2  +  px  =  q  may  have  real  roots. 

13.  Do  both  roots  of  a  quadratic  equation  necessarily  satisfy 
the  conditions  of  the  problem  from  which  the  equation  may  be 
derived  ?     How  can  such  results  be  checked  ? 

REVIEW  EXERCISE 

549.  Solve: 
x     2-5x      1+x  148  -5a2      _2 

5x+l     3-2x     3  +  13a;-10a;2 

2.    -^—  +  ^^=25. 


3.  J?0+9_j20T9  =  a 

*x2       Vx2 

4.  (a>+ll)*  +  3(»  +  ll)*  =  4. 

u         1  x  -h  a  x2 

5. 


x  +  a      x  —  a     a2  —  x2 


6.    Vz+18+Va?-18  =  6. 

7     °  ~  a     a  —  253  #(q  —  b)  _  ft 
'a;  —  6       a;  +  6  x2  —  b2 


8.  a:2  +  12  a  =  2  Va?2  +  12  a?  -  4  +  67. 

9.  ^-1=1^. 

10.  Give  nature,  sum,  and  products  of  the   roots   of  the 
following  : 

(a)  5x2-7x  +  2  =  0.  (d)   2x2-6x  =  -m,  whenra  >  4|. 

(6)  a2-5=4a>.  (e)   5-  3a*  +  7a;  =  0. 

(c)  a*  +  2  =  0.  (/)    3a;2-3a;  +  |=0. 


398  Quadratic  Equations 

11.  Form  equations  whose  roots  are : 

(a)  7,  3.  (d)    a  —  b,  b  —  a. 

(&)  h-h  00    3±V2. 

(<0  3,f  (/)    3±«. 


12.    Solve7a+Va2-17x  +  4  =  2a;2-27a;  +  5. 

«  Q     «  -+■  Vl2  a  —  x      Va  +  1 
#  —  Vl2a  —  x      Va  — 1 

14.  For  what  value  of  m  are  the  roots  equal  in 

Sxi-bx  +m  =  0? 

15.  What  change  is  made  in  the  roots  of  the  complete 
quadratic  by  changing  the  sign  of  b  ? 

16.  a;  +  l--^==-^  +  5a?-4- 

1  -  X2      x-\-l       x2-l 

17.  In  two  years  the  population  of  a  city  increased  from 
6400  to  8100 ;  the  rate  per  cent  of  increase  during  the  first 
year  was  equal  to  the  rate  per  cent  of  increase  during  the 
second  year.     What  was  this  rate  ? 

18.  In  the  equation  x2  +  y2  =  1,  y  =  ^  V3  ;  find  x. 

19.  Solve,  getting  the  answers  correct  to  two  decimal  places, 

3a;2_a._-v/2  =  0. 

20.  Solvea2+a+3a;V3  +  4  =  0.     (Yale.) 
One  answer  is  1  —  V3. 

21.  A  pedestrian  having  18  miles  to  go  to  keep  an  appoint- 
ment finds  that  at  his  present  rate  he  will  be  half  an  hour  late. 
If  he  quickens  his  pace  by  half  a  mile  an  hour,  he  will  arrive 
on  time.     At  what  rate  is  he  walking  ? 

22.  A  room  is  two  yards  longer  than  it  is  wide  and  the 
floor  contains  24  square  yards.  Find  the  dimensions  of  the 
room. 


Solve  : 


Review  Exercise  399 


23.    12^     37-3s 

x  -  5  T  25  -  a* 

a;        .  3a;-2 


24.    = I-  ^-^ =  =  4-. 

2(z+2)^  4 -a*       2 


25.  V5a?4-  1  —  V3#  =  l. 

26.  V2<e  — 1  +  V2a  +  6  =  7. 

27.  Va  +  3  +  Va;  +  8  =  5Va;. 


28.    V3aj-f  1+V5a;  +  4  =  3. 


29.  V5a;  +  10-V5z  =  2. 

30.  6a#-lla^-2  =  0. 

31.  V3ic-2+6  =  5\/3a;-2. 

32.  — *— -1 5=0. 

2*-l      V2a>-1 

33.   Find  a  number  such  that  one  half  its  square  shall  ex- 
ceed the  square  of  one  half  the  number  by  one  half  the  number. 


34.    — z== =     •      (Apply  composition  and  division.) 

Va*+1-Va*-1      6 

x  —  -Vox  .  a  —  VaaT  _  a;  —  a  ^        (Clear  of  fractions  and  fac- 


35. 

a  +  y/ax     x-\-^/ax         a  t/01^ 


XXHL     QUADRATIC  EQUATIONS  (Continued) 
SIMULTANEOUS   EQUATIONS 

550.  The  degree  of  an  equation  is  determined  with  respect 
to  the  letters  that  are  regarded  as  unknowns.  For  example, 
ax  +  b  =  0  is  of  the  first  degree  with  respect  to  x  as  unknown. 
(§§  245,  246.) 

s  =  \gff-  is  of  the  second  degree  with  respect  to  t  as  unknown. 
ax  +  by  =  c  is  of  the  first  degree  with  respect  to  x  and  y. 
3xy  +  x  +  y  =  5  is  of  the  second  degree,  regarding  x  and  y 
as  unknowns. 

551.  There  is  no  general  method  for  solving  simultaneous 
quadratic  systems  since,  in  general,  the  elimination  of  one  of 
the  unknowns  gives  rise  to  an  equation  of  higher  degree  that 
cannot  be  solved  by  methods  of  elementary  algebra.  We  shall 
consider  some  special  forms  of  quadratic  systems. 

552.  Case  I.  One  Equation  of  the  First  Degree  and  the  other 
of  the  Second.     (Review  of  Chapter  XVII.) 

A  system  of  equations  involving  two  unknowns,  one  equa- 
tion linear  and  the  other  quadratic,  can  always  be  solved,  since 
the  elimination  of  one  unknown  by  substitution  from  the  first 
degree  equation  into  the  second  degree  equation  gives  rise  to  a 
quadratic.     (See  §  430.) 

Solve  the  system  (x  +  y)(x  —  2  y)  =  7.  (1) 

x-y  =  3.  (2) 

Solution.  x  =  y  +  3.     (From  (2).) 

(V  +  3  +  y)  (y  +  3  -  2  y)  =  7.     (Substituting  in  (1).) 
(2y  +  8)(-y  +  8)  =  7. 

400 


Simultaneous  Equations  401 

2y2_3y_2  =  0.        

y  =  3±V9  +  16=2and_l, 

When  y  =  2,^  =  2  +  3  =  5.       4  2 

When    |f  =  -|,«=-i  +  8s=|. 

x  =  5,  f . 

V  =  2,  -  J. 
Check  both  sets  of  roots. 

EXERCISE 

553.  #oZve  Me  following  systems,  grouping  the  answers  properly 
at  the  end  of  the  solution.  Leave  irrational  answers  in  simplest 
radical  form.     Verify  one  set  of  answers  in  examples  1  to  6: 


1. 

x  +  y  =  13, 

11. 

ab  =  147, 

icy  =  36. 

a  :  6  =  3  : 1. 

2. 

a?  +  y  =  10, 
a;2  +  y2  =  58. 

12. 

M=5, 

x     y 

3. 
4. 

ay  —  5  a?  =a  1, 

7x  —  y  =  l. 
x2  +  4:xy  =  57, 
*  +  y  =  7. 

Hint.     Use  -    and    -    as    un- 

.                    *          y 

knowns. 

5. 

3#  +  5y  =  35, 

13. 

x  +  y  =  29, 

aj2+2  y2=a.y+8  a>— y+13. 

V#  +  Vy  =  l. 

6. 

7. 

5x  —  y_        7 

HlN 

known 
14. 

t.    Use  Vx  and  -\A/  as  un- 

4      ~~4<c  +  3^ 
8»-2yal 

4     5 

(r  +  sy  =  200  -  r. 

s. 
x     y     2 
x2     y2     4 

8. 

3  m  —  n  =  5, 

15. 

x  +  y  =  a, 

wm  —  m  =  0. 

x2  -\-  y2  =  2xy. 

9. 

2*5=1. 

16. 

p2=W, 
Sp  =  5q. 

10. 

x2  +  2/2  =  130, 
*  +  .V=8. 

17. 

0     *      3' 

»  —  2/  4:X—7y  =  5. 


402  Quadratic  Equations 

Solve  the  following  systems: 

Note.  It  frequently  happens  that  other  systems  can  be  reduced  so  as 
to  consist  of  one  equation  of  the  first  degree  and  one  of  the  second.  In 
example  18  eliminate  y2  by  adding  the  equations. 

18.    x  +  y  +  2tf  =  ll, 
Sx-2y-2y2  =  -9. 

19.  xy-x  =  12,  21.   S(x2  -  y2)  =  2  x  +  17, 
xy  +  3  y  =  35.  x2  -  y2  +  x  +  y  =  18. 

20.  (w  +  l)(v  +  2)=28,  22.   4a;2/-5ic2-2a?  +  2/  =  12, 
(w  +  3)(v  +  4)  =  54.  x(4:  y  —  5  x)  =  13  —  x  +  y. 

Note.  The  method  of  elimination  by  substitution  will  sometimes 
solve  a  system  when  one  equation  is  of  the  third  degree,  and  the  other  of 
the  first  degree. 

23.  ar5  +  ?/3  =  9,  25.   v3  -  w3  =  1304, 
x  +  y  =  3.  v  —  ?4  =  8. 

24.  ^-^  =  19,  26.    ^  +  ^  =  65, 
x  —  y  =  l.  t  +  r  =  5. 

27.    «2  +  2/2  +  ^y  =  67, 
x  +  y  =  9. 

554.  Case  II.     Homogeneous  Equations  of  the  Second  Degree. 

An  algebraic  expression,  or  an  equation,  in  which  all  the  terms 
are  of  the  same  degree  is  homogeneous. 

The  homogeneity  of  a  literal  equation  is  determined  with 
respect  to  the  unknown  numbers. 

Thus,  a3  +  3  a2b  is  a  homogeneous  expression. 
ax2  +  5  xy  =  0  is  a  homogeneous  equation  in  x  and  y. 
x2  +  5  xy  +  x  +  3  y  =  10  is  not  homogeneous.     (Why  not  ?) 
x2  +  3  xy  +  y2  =  5,  is  homogeneous,  except  with  respect  to  the  absolute 
term. 

555.  It  is  always  possible  to  solve  a  simultaneous  system 
when  both  equations  are  of  the  second  degree  and  one  of  them 
is  homogeneous. 


Simultaneous  Equations  403 

Solve  2x2  +  xy-6y2  =  0, 

3x*-4:xy  =  3. 

Solution.       (2  x  —  3  y)(x  +  2  y)  =  0.     (Factoring  the  first  equation.) 
The  first  equation  is  therefore  equivalent  to  the  two  equations,  2  x  — 

3  y  =  0  and  x  +  2  y  =  0.     We  now  form  two  systems,  using  each  of  these 

first  degree  equations  with  3  x2  —  4  xy  =  3. 


System  A.     x  +  2y  =  0. 

System 

i  B.     2  x  -  3  y  =  0. 

3  x2  -  4  xy  =  3. 

3  x2  -  4  xy  =  3. 

2 

*       3 

3x2-4x(-?)=3, 

3x2-4x-  2^  =  3 
3 

or  3  x2  +  2  x2  =  3. 

x2  =  f . 

or3x2-^-2  =  3. 
3 

x2  =  9. 

5 

x  =  ±3, 

*          2           10 

x  =  ±3, 

± 

Vl5# 
5 

andy=^  =  ±2. 
o 

y  =  ±2, 

=F 

Vl6 
10  ' 

When  signs  are  paired  in  this  way  it  is  understood  that  the 
top  signs  are  to  be  used  together  and  the  bottom  signs 
together. 

Vi5    -Vis 


Thus,  x  =  3,  -3, 

V  =  2,  -  2, 


5    '       5 
-\/l5     V15 


10     '     10 


Let  the  student  check  the  answers,  substituting  them  in 
the  original  equation. 

556.  To  solve  a  system  of  two  second  degree  simultaneous  equations 
when  one  is  homogeneous  : 

1.  Write  the  homogeneous  equation  with  its  second  member  zero,  and 
factor  the  first  member. 

2.  Put  each  factor  equal  to  zero  (why?)  and  use  each  of  the  two 
linear  equations  obtained  with  the  other  one  of  the  two  original  equa- 
tions as  in  Case  I. 


404  Quadratic  Equations 

Solve  the  system     3  x2  —  8  xy  +  4  y2  =  0, 
x2  +  y2  +  13(<c  -  y)  =  0. 

Solution.     (3  x  —  2  y)  (x  —  2  y)  =  0.     (Factoring  the  first  equation.) 
System  A.     3x  —  2y  =  0,  System  B.     x  -  2  y  =  0, 

x2  +  y2  +  13(x  -y)  =  0.  x2  +  2/2  +  130  -  y)  =  0. 

3-tU.  »  =  2y. 

3  (2  2/)2  +  2/2+13(2?/-?/)=0, 

Uy        ^        A3       yJ  y(5jf  +  13)=0. 

or13_^_13y  =  0  y  =  0or-¥, 

9  3  and  x  =  2j/  =  0or-3&. 

2/2-32/ =0. 

y  =  0  or  3, 
and  x  =  §  t/  =  0  or  2. 
a?  =0,2,  0,  -y, 
and  J/  =  0,  3,  0,  -  J£. 

557.  If  both  equations  are  of  the  second  degree  and  homo- 
geneous except  with  respect  to  the  absolute  term,  the  absolute 
term  may  be  eliminated   and  the  method  of  the  last   article 

applied.  _      A 

rxr  x2  —  3  xy  =  4, 

3  x2  +  xy  —  2  y2  =  50. 

Solution.     25  x2— 75  x?/=100.     (The  first  equation  multiplied  by  25.) 

6  x2  +  2  xy  —  4  2/2  =  100.  (The  second  equation  multiplied  by  2.) 

19  x2  -  77  xy  +  4  y2  =  0.  (Subtracting. ) 
(19x-?/)O-42/)  =  0. 

System  A.    y  =  19  x,  System  5.     x  =  4  y, 

x2  —  3  X2/  =  4.  x2  —  3  xy  =  4. 

x2  -  57  x2  =  4.  16  j/2  -  12  i/2  =  4. 

56  x2  =  -  4.  4  2/2  =  4. 

x2=-^.  2/2  =  L 

x  =  ±AVl4,  ,         /  =  ±I' 

14  and  x  =  4  y  =  ±  4. 

and2/=19x  =  ±^Vl4. 

The  answers  are  x  =  ±  4,  ±  —■  Vli. 
14 

14 


Simultaneous  Equations  405 

The  student  will  note'  that  the  elimination  of  the  absolute 
terms  of  the  original  equations  is  similar  to  the  elimination  of 
one  of  the  unknown  numbers  in  a  system  of  linear  simultane- 
ous equations  by  the  method  of  addition  and  subtraction. 

558.  Optional  Method.  When  the  equations  are  homogeneous 
and  of  the  second  degree,  the  substitution  oiy  =  vx  will  always 
effect  a  solution.  The  solution  of  the  last  example  by  this 
method  is  shown  below  : 

Solution.  x2  —  3  xy  =  4. 

3  x2  +  xy  —  2  y2  =  50. 

Substitute  vx  for  y  in  both  equations. 

4 

x2  —  3  vx2  =  4,  whence  x2  = 

l-3« 

3  x2  +  vx2  —  2  v2x2  =  50,  whence  x2  = 

4  50 

1-SV     S-\-v-2v2' 
12  +  4v-8v2  =  50  -  150  v. 
-  8v2  +  154  v  -  38  =  0. 

4  v-2  _  77  v  +  19  =  o. 


(i) 

(2) 

(From  (1).) 

50 

(From  (2).) 

3  +  v  -  2  v2 

(Equating  the  two  values  of 

a;2.) 

r=77±\/772-  4-  4. 19 

=  77±75  =  819orl. 
8  4 

System  A.     v  =  19.  System  B.    v  =  \. 


1-Zv 
4  1 


3v 


356  =  ~iT  #-|«ia 

19 1 
and  y  =  tw;  =  ± -^  Vl4.  and  y  =  vx  =  £(±  4)  =  ±  1. 

4ns.    s  =  ±-^-Vl4,  ±4, 
3/=±^VT4,   ±1. 

By  comparing  the  two  methods  it  will  be  seen  that  there 
is  not  much  difference  in  the  amount  of  work  involved. 


406  Quadratic  Equations 


EXERCISE 

559.   Solve  the  following  systems.     Verify  one  set  of  answers 
in  each  system  in  examples  1  to  6  : 

1.  x2-  42/2=0,                           13.  x2-3xy  =  5xy-15y2, 
x2  +  3xy  +  y  =  24:.  x2-3xy  =  10. 

2.  15x*-16xy-15y*=0,       14.  a*  +  *2  =  661, 

3x2  +  5y2  =  120.  a* -2?  =589. 

3.  3x2-±xy-7y2  =  0,           15<  xi  +  xy  +  y2  =  7% 
5^  =  42/2-1.  x2-xy+y2  =  37. 

4.  (3aj-y)(3»  +  y)=35, 

5  g2  _  4^g  =  _  75. 

5.  x2  —  xy  =  5, 

a;2/  +  y2  =  36.  W-  2  a2  -  ax/ =  28, 

a;2  +2  2/2  =  18. 

6.  x2  +  3xy  =  5, 

xy-y2  =  6.  18.  (s  -*)(«+  *)  =  40, 

2/2 +  6x2/ =  13.  19>  x2+  32/2  =  7, 

a;^?/     5  7a;2-5x'2/  =  18. 

o.     -  +  -  =  -, 

V     ®     *  20.  «2  +  a;2/  =  a, 

^  =  8-  if  +  Xy=  b. 

9.   ^-±1  +  ^^  =  5,  21.  aa;2  4-6a?2/=a, 

aj2+2a;2/-32/2  =  5.  /aj       0^-18. 

11.  (x  +  y)2=3x2-2,                 23.  3z2  +  3xy  +  2  2/2  =  8, 
(x  -  2/)2  =  3  2/2-  11.  x*  -  ^2/  -  42/2  =  2. 

12,  (2aj+5.y)(3aj-5y)=44,       24.  6 a2  +  5 ay  -  6 2/2  =  0, 
#2_Qxy+  122/2  =  3.  2a?2-2/2  =  -l. 


Simultaneous  Equations  407 

560.  Case  III.  Symmetrical  Equations.  An  equation  is 
symmetrical  if  an  interchange  of  the  unknowns  does  not 
change  the  equation  except  in  the  order  of  its  terms. 

Thus,  x  -f-  3  xy  -j-  y  =  10  becomes,  by  interchanging  x  and  y, 
y  _j_  3  yX  +  x=.  10.     Therefore  the  equation  is  symmetrical. 
x  +  y  =  5  becomes  y  -f-  x  =  5  and  is  therefore  symmetrical. 
Is  x2  +  y2  =  5  symmetrical  ?     a;#  +  a;  =  10  ?     ic  +  2  ?/  =  5  ? 

561.  A  system  consisting  of  two  symmetrical  'equations  can 
generally  be  solved  by  combining  the  equations  in  such  a  way 
as  to  find  values  of  x  -f-  y  and  x  —  y. 

Solve  the  system,         x2  +  y2  =  17,  (1 ) 

x+y  =  5/  (2) 

Solution.  Here  we  have  the  value  of  x  +  y ;  we  look  for  zy  and 
thence  x  —  y. 

x2  +  2xy  +  y2  =  25.     (Squaring  equation  (2)).  (3) 

2  xy  —  8.    (Subtracting  equation  (1 )  from  equation  (3)).   (4) 
x2  —  2  ojy  +  y2  =  9.     (Subtracting  equation  (4)  from  equation  (1)). 
B-y=±8, 
We  now  replace  the  original  system  by  the  two  systems  : 
A.    .        x  -f  y  =  5.  B.  x  +  y  —  5.  ^4ns.  a;  =  4,  1. 

x-y  =  3.  £—  y=-3.  y=l,  4. 

Whence  x  =  4.  Whence  #  =  1 . 

y  =  l.  2/ =  4. 

562.  1.  The  student  should  follow  some  such  systematic 
arrangement  of  his  work  as  is  found  in  §  561. 

2.  He  should  carefully  study  the  equations  to  determine 
what  steps  will  lead  to  the  desired  forms.  No  general  rules 
can  be  given  since  the  method  of  procedure  varies  with  the 
form  of  the  equations. 

3.  It  will  usually  be  found  helpful  to  find  a  value  of  xy,  as 
in  (4)  §  561,  and  use  this  value  in  combination  with  one  of 
the  preceding  equations  to  form  an  equation  containing  some 
power  of  x  +  y  or  x  —  y. 

4.  From  this  equation  values  of  x+y  and  x—y  can  be  found. 


408  Quadratic  Equations 

5.  It  will  be  well  also  to  divide  the  equations,  member  by- 
member,  when  this  is  found  possible.  Thus,  if  Xs  -f  yz  =  15  and 
x  4-  y  =  3,  by  dividing  we  get  x2  —  xy  -f-  y2  =  5. 

563.  Optional  Method.  Symmetrical  equations  can  be  solved 
also  by  the  following  method: 

a*  +  y2  =  17,  (1) 

x  H-  y  =  5.  (2) 

Substitute  w  +  -y  for  x  and  w  —  v  for  ?/. 

Equation  (1)  becomes  (u  +  v)2  +  (w  -  v)2  =  17.     (From  (1).)  (3) 

2m2  +  2v2  =  17.  (4) 

(w  +  «)  +  («-«)=  5.       (From  (2).) 
2m  =  5. 
w  =  f. 
2(|)2  +  2v2  =  17.     (From  (4).) 

o2  =  f . 

jc  =  M  +  i?  =  |±f  =  4orl, 
and  y  =  u  —  ^  =  ^  +  1  =  1  or  4. 

EXERCISE 

564.  Solve  the  following  systems,  and  verify  one  set  of  answers 
in  examples  1  to  5 : 

1#    #2  _j_  <^2  _  5Q?  Multiply  the  second  equation  by  2  and  com- 

Xy  _  j>  bine  with  the  first  equation  to  get  values  of 

x2  +  2  xy  +  I/2  and  a;2  —  2  a?/  +  y2. 

2.  a2  —  xy  +  ?/2  =  7,  6.    x2  +  a#  +  y2  =  6, 

SB  +  ?/  =  4.  x2  —  xy  +  y2  =  —  6. 

3.  a2 +  #2/ +  y2  =  14,  7.   x2  +  ?/2 +  »  +  i/  =  146, 
x  —  y  —  V2.  #2/  =  63. 

4.  x3  +  2/3  =  98,  8.   a3  +  2/3  =  28, 
a;2  —  a^/  -f-  y2  =  49.  a;  +  y  =  4. 

5.  x2  +  */2  =  269,  9.   a?-#3  =  26, 
a;  —  y  =  3.  a;  —  y  =  2. 


Simultaneous  Equations  409 

10.    (x-y)2-2>(x-y)  =  ±, 
x2  +  y2  +  2  xy  =  49. 

Solution.     From  the  first  equation  x  —  y  =  4  or  —  1.     (Why  ?) 
From  the  second  equation  x  +  y  =  ±  7.     (Why  ?) 

We  have,  then,  the  four  systems. 

A.  x  +  y  =  7,       B.x  +  y  =  7,       C.x  +  y=-l,       D.x  +  y  =  -7, 
x  —  y  =  4.  x  —  y  =—  1.         £  —  y  =  4.  a;  —  y  =  —  1. 

Let  the  student  complete  the  solution. 

11.   a*+0*-(a>+y)-12=O,  12.   3(aj»+2^)  =  8(a?  +  y)-l, 

ajy-2(s  +  y)+8  =  0.  ay  =  (x  +  y)+l. 

13.   x2  -\-  y2  =  xy  =  x  -{-  y. 

x2  +  2xy  +  y2  =  x2y2.     (From  x-\-y  =  xy.) 
But  z2  +  y2  =  xy. 

2  zy  =  x2y2  —  xy.    (Subtracting.) 

Let  the  student  find  the  values  of  xy  in  the  last  equation  and  continue 
the  solution. 

14.   x4  +  a22/2  +  ?/4  =  21.  i5.   x2  +  xy  +  y2  =  91, 

a?2  +  a*/  H-  ?/2  =  7.  x-\- -\/xy  +  y  =  7. 

Divide  the  first  equation  by  the  second. 


16. 

&  +  y3  =  —  2  xy, 

x  +  y  =  -2. 

21. 

p+pq  +  q  =  ±7, 
P  +  q  =  12. 

17. 

x*  +  y3  =  280, 
x2  -  xy  +  y2  =  28. 

22. 

x     y      2 

18. 

x2  +  y2  +  x  +  y  =  168, 

^Jxy  =  6. 

I+i=A. 

z2     ?/2     36 

19. 

x2  +  2/2  +  a  +  y  =  18, 
2  a#  =  12. 

23. 

r2  +  rs  -f-  s2  =  21' 
r  +  i»  17. 

20. 

ar  =  -jVz  +  2/, 
y  =  ±£^/x  +  y. 

24. 

a?4  -  t  =  6°9, 

a2  +  y2  =  203. 

410  Quadratic  Equations 

GENERAL  SUGGESTIONS  FOR  THE   SOLUTION  OF  SIMUL- 
TANEOUS QUADRATIC  EQUATIONS 

565.  In  solving  a  quadratic  system,  first  determine  under 
which  one  of  the  following  three  cases  it  occurs : 

I.  One  equation  of  the  first  degree  and  the  other  of  the  second. 

II.  Homogeneous  second  degree  equations. 

III.  Symmetrical  systems. 

If  the  system  does  not  come  under  one  of  these  three  cases, 
try  to  derive  another  system,  or  other  systems,  from  the  given 
equations  that  will  come  under  one  of  these  three  cases.  At 
all  times  remember  that  the  object  is  to  eliminate  one  of  the 
unknowns. 

566.  Among  special  devices  may  be  mentioned  the  following : 

1.  The  immediate  elimination  of  one  of  the  unknowns  by 
addition,  subtraction,  or  substitution. 

2.  The  elimination  of  the  second  degree  terms. 

3.  Finding  a  quadratic  form  in  some  expression  containing 
the  unknowns  and  solving  for  the  value  of  this  expression.  If 
this  expression  is  of  the  first  degree  in  the  unknown,  the  given 
system  may  be  replaced  by  two  systems  under  Case  I. 

4.  Dividing  the  equations  member  by  member. 

EXERCISE 

567.  Solve  the  following  systems : 

1.    x:y  =  2:S,  5.    ox  -  lOy  +  (x  -  2y)2=  6, 

x2  +  y2  =  5{x  +  y)—  2.  xy  +  x+y  =  7. 


2. 

O  +  2/)2+0»+2/)=12, 

6.    ^  +  5^  =  14, 

y2      y 

3  x2  +  i/2  =  x  -f-  y  +  4. 

3. 

x2  +  y2  =  34, 
xz-2y2  +  3x  =  -50. 

x  =  y2  +  l. 
7.   5^+3^=8, 

4. 

x(y-4)=U, 

y      * 

y(a>  +  l)  =  33. 

x2  +  y  =  x  -f-  4. 

Simultaneous  Equations  411 


8.  4:X2-9xy  +  5y*  =  0, 
7  x2—  3xy  =  3x  +  2y  —  l. 

9.  2x2-3y*  =  6, 
3a? -2/  =  19. 

10.  (x+y)(x-2y)=7, 
x-y  =  3. 

11.  — - r^-  =  a, 
1-xy 

x  =  y. 

15.  3xy-2(x-\-y)=2Sf 
2xy-3(x  +  y)=2. 

16.  (2x-2/)2-12(2»-v/)  =  189, 
x2  —  4  xy  +  4 1/2  —  3  x  +  6  ?/  =  54. 

17.    z2  +  2/2-2(x-2/)  =  38,  23.   4:X2-9y2  =  0, 

a^  +  3(a;~y)=25.  4  a;2  +  9  y2  =  S(x  +  y). 


12. 

*  +  y  _i 
1  —  ay 

1 
a;  = 

V3 

13. 

»2  +  2/2  +  ^+y  =  18, 

#2  —  #2  -f  x  —  .V  =  6. 

14. 

a?  _  ^  =  40, 

a*/  =  21. 

24.  a;2  +  2/2  =  a^  +  189, 
60(a;—  2/)  =  xy. 

25.  ^+^  =  3, 


1  —  ay 

19.    a/l+?La»  g~y  =1 

V        y/  l  +  ajy     3 


y(i  + 


x 


26.    x2-a*/  =  3, 
xy-y2  =  2. 


20     _  _i_  _  =  5  

'    x     y        '  27.    2a;+Va#=10, 

a;  — ^=.3.  3?/—  2yjxy  =  —  1. 


21.  Va?  —  5  +  \A/  +  2  =  5, 
x  +  ?/  =  16.  ^  4-  6a?  +  c  =  0. 

22.  8(z-5)2-3(2/-7)2=80,  29.   x2  +  ?/2  +  a?  +  2/  =  36, 
4(a-5)2+  5(2/-  7)2=  144.  2(x2  +  y2)  +  3  ay  =  88. 


412  Quadratic  Equations 

30.   x2  +•  y2  -  xy  =  7,  32.   x2y  +  xy2  =  30, 

a;     ?/     6 


(a>  — y)+a#=5.  1.1-5. 


31.   x2  +  »2/-«  =  0, 

2/2  +  xy  —  6  =  0.  33.    x2y2  +  Xy  —  2=0, 

x+y  =  —  l. 


to  lor  -f. 

35. 

rc+2/+z  =  2, 

xy  =  -l, 

xyz  =  —  2. 

36. 

ic?y  =  12, 

#z  =  15, 

?/2  =  20. 

37. 

a-f-y-f  z  =  6, 

2a-2/+z=3, 

a.2  +  ^2_|_^==  14 

34.    If  x2  —  2  y  =  —  J  and  ?/  =  Vl  —  x2,  show   that  #   is  equal 


38.  xy  -\-xz  —  80, 
xz  -\-yz  =  108. 

39.  a?  +  y  +  2!  =  37, 
a2  +  2/2+z2  =  481, 
i/2  =  axs. 

40.  (z  +  l)(y  +  l)  =  15, 

(jr  +  l)(s  +  l)«3* 
(«  +  l)(«  +  l)-21. 

ELIMINATION 

568.  It  often  happens  in  the  study  of  mathematics  and 
physics  that  it  is  necessary  to  eliminate  one  or  more 
unknown  quantities,  or  variable  quantities,  and  either  solve 
the  resulting  equation  for  the  other  unknown  or  derive 
an  equation  containing  it.  Some  one  of  the  methods  of  the 
present  chapter  will  generally  accomplish  the  desired  elimi- 
nation. 

W=fs,  f=  ma,  s  =  i  at2,  v  =  at  Find  W  in  terms  of  m 
and  v ;  that  is,  eliminate  /,  s,  a,  and  t. 

Solution.     W  =  ma  •  \at<i    Substituting  for  /  and  s. 
=  \  maH2.       Why  ? 
.-.  W=\mv2.  Why?    , 


Elimination  413 

EXERCISE 

1.  Given  v  =  gt  and  s  =  ^gP;  eliminate  t  and  show  that 
v  =  V2  gs.     (Physics.) 

2.  Given   vl  =  vQ  -f  kvQtx  and  v2  =  v0  -f-  kv0t2 ;   eliminate  vQ. 
(See  §  566,  4.)     (Physics.) 

3.  Solve  the  result  in  problem  2  for  k. 

or    I    ?/  2  'i* 

4.  Given  — -±-2-  =  1  and  y  = ;  eliminate  y. 

1  —  xy  1—x2 

5.  #  =  -(a  +  Z)   and   l  =  a+(n—  l)d.     Eliminate   I,  and 
express  the  value  of  S  in  terms  of  a,  n,  and  d. 

6.  «;?/  +  zx  =  1, 

w  =  2  a?Vl  -  a2, 

y  =  Vl  —  a?, 

z  =  y2—  x2. 
Eliminate  w,  y,  2;. 

The  resulting  equation  in  x  should  be  satisfied  by  x  =  \. 

7.  Given  (7  =  —  and  C"  = ;  eliminate  j£.     Also  find 

R  R  +  r' 

E  in  terms  of  C,  C",  and  r. 

8.  If  v  =\l-  and  k  = — 3L,  find  the  value  of  v  after  elimi- 

yid  irR2' 

nating  k. 

9.  Given  C=     *  +  A        and  C  =     ^  ~  ^      ;   find  #' 

/£  +  Jff>  +  r  iJ  +  U'  +  r' 

after  eliminating  i2  +  i£'  +  r. 

10.  Given   v  =  u  —  gt  and  s  =  ut  —  \g&\    eliminate   £  and 
solve  the  resulting  equation  for  w. 

11.  Given  (7= and  C  =  — ;  eliminate  r. 

R  +  r  R  +  nr 

12.  Eliminate  x  from  the   two  equations  ax2  +  bx  +  c  =  0, 
and  2  a#  +  5  =  0. 

13.  Given  F  =  —  m  and  —  =  i;  show  that  F=  BIL. 

d2      '         m      d2 


414  Quadratic  Equations 

PROBLEMS    LEADING    TO    SIMULTANEOUS    QUADRATIC    EQUATIONS 

569.  1.  The  difference  of  the  two  sides  about  the  right 
angle  in  a  right-angled  triangle  is  2  inches.  The  hypotenuse  is 
10  inches  long.     Find  the  unknown  sides. 

2.  The  sum  of  two  sides  about  the  right  angle  in  a  right 
triangle  is  s  and  the  hypotenuse  is  h.  Express  the  values  of 
the  two  sides  about  the  right  angle  in  terms  of  s  and  hi 

3.  The  difference  between  the  sides  and  a  diagonal  of  a 
square  is  3  inches.  Find  the  side  and  diagonal  to  two  decimal 
places. 

4.  The  sum  of  the  roots  of  a  quadratic  equation  is  2,  and 
their  product  is  —  1.     Find  the  roots  and  make  the  equation. 

5.  If  a   polygon   has  n  sides,  it  has  - — }-  diagonals.  I 

Two  polygons  have  together  18  sides,  while  the  number  of 
diagonals  of  the  one  is  to  the  number  of  diagonals  of  the  other 
as  4  to  7.     How  many  sides  has  each  ? 

6.  Two  polygons  have  together  12  sides  and  19  diagonals. 
How  many  sides  has  each  ? 

7.  Two  adjacent  square  plots  of  unequal  sides  are  inclosed 
by  a  continuous  fence.  The  total  area  of  the  fields  is  5200 
square  rods  and  the  length  of  the  fence  is  1  mile.  How  large 
is  each  plot  ?     (Draw  a  figure.) 

8.  Besides  zero  there  are  two  pairs  of  numbers  such  that 
their  sum,  their  product,  and  the  difference  of  their  squares 
have  the  same  value.     Find  these  numbers. 

9.  In  a  proportion  the  sum  of  the  means  is  5  and  the  sum 
of  the  extremes  is  7.  The  sum  of  the  squares  of  all  the  terms 
is  50.     Find  the  terms.     (Use  only  2  unknowns.) 

10.   Find  two  factors  of  p  whose  sum  is  s. 


Problems 


415 


11.  In  the  figure,  CD  forms  a 
right  angle  with  AB.  The  sides 
are  9  inches,  12  inches,  and  14 
inches  as  indicated.  Find  the 
length  of  x  and  y  to  one  decimal 
place. 

(Note  that  this  would  enable  us  to  find  the  area  of  a  triangle  when  we 
have  given  the  three  sides. ) 

12.  A  mean  proportional  between  two  numbers  equals  VlO. 
The  sum  of  the  squares  of  the  numbers  is  29.  Find  the 
numbers. 

13.  The  sum  of  the  areas  of  two  squares  is  125  square 
inches.  The  sum  of  their  four  diagonals  is  30V2  inches. 
Find  the  sides  of  the  squares. 

14.  Find  the  dimensions  of  a  rectangular  room,  knowing 
that  the  floor  has  an  area  of  240  square  feet ;  one  side  wall 
contains  180  square  feet,  and  one  end  wall  108  square  feet. 

15.  The  sum  of  two  numbers  is  one  sixth  of  the  difference 
of  their  squares,  and  the  sum  of  the  squares  is  306.  Find  the 
numbers. 

16.  Divide  84  in  two  parts  such  that  the  sum  of  their 
squares  is  3560. 

17.  If  to  the  product  of  two  numbers  is  added  the  greater 
number,  we  obtain  855  ;  but  if  to  the  same  product  is  added 
the  smaller  number,  we  obtain  828.     Find  the  two  numbers. 

18.  The  sum  of  the  squares  of  two  numbers  is  410.  If  the 
greater  number  is  diminished  by  4  and  the  smaller  number  is 
increased  by  4,  the  sum  of  the  squares  is  394.  Find  the  two 
numbers. 

19.  A  number  is  formed  of  two  figures  of  which  the  sum  is 
13.  If  34  is  added  to  the  product  of  the  two  figures,  the  sum 
is  equal  to  the  number  obtained  by  reversing  the  figures  of  the 
first  number.     Find  the  number. 


416  Quadratic  Equations 

20.  The  diagonal  of  a  rectangle  is  65  inches.  If  9  inches  are 
added  to  the  width  and  3  inches  subtracted  from  the  length  of 
the  rectangle,  the  diagonal  remains  the  same.  Find  the  di- 
mensions of  the  rectangle. 

21.  The  diagonal  of  a  rectangle  is  89  feet.  If  each  side  of 
the  rectangle  is  diminished  by  3  feet,  the  diagonal  will  be  85 
feet.     Find  the  length  of  each  side. 

22.  The  hypotenuse  of  a  right-angled  triangle  is  35  feet.  If 
the  shorter  side  is  diminished  5  feet  and  the  longer  side  in- 
creased 2  feet,  the  hypotenuse  will  be  1  foot  less.  Find  the 
two  sides  of  the  triangle. 

23.  Two  square  gardens  have  a  total  area  of  2137  square 
yards.  A  rectangular  lawn  of  which  the  dimensions  are  equal 
respectively  to  the  sides  of  the  two  squares  has  an  area  of 
1093  square  yards  less  than  that  of  the  two  gardens  together. 
Find  the  sides  of  the  two  squares. 

24.  The  sum  of  the  areas  of  two  circles  is  13,273.26  square 
inches,  and  the  sum  of  their  radii  is  79  inches.  Find  the  two 
radii. 

25.  The  sum  of  the  surfaces  of  two  spheres  is  1000  square 
inches,  and  the  sum  of  the  radii  is  12  inches.  Find  the  two 
radii  correct  to  two  decimal  places.  (The  surface  of  a  sphere 
is  AvB2  square  inches  if  R  is  the  number  of  inches  in  radius.) 

26.  The  sum  of  the  surfaces  of  two  spheres  is  14,388.53 
square  inches,  and  the  difference  of  the  radii  is  9  inches.  Find 
the  radii. 

27.  The  sum  of  the  volumes  of  two  spheres  is  14,778.0864 
cubic  inches  and  the  sum  of  the  radii  is  20  inches.  Find  the 
radii  correct  to  two  decimal  places.  (The  volume  of  a  sphere 
equals  |  ttRs  cubic  inches  if  R  is  the  number  of  inches  in  the 
radius.) 

28.  Three  numbers  are  such  that  if  we  take  the  product  of 
them  two  at  a  time  the  results  will  be  240,  160,  and  96.  What 
are  the  numbers  ? 


Problems  417 

29.  The  sum  of  three  sides  of  a  right-angled  triangle  is 
208  feet.  The  sum  of  the  two  sides  about  the  right  angle  is 
30  feet  longer  than  the  hypotenuse.  Find  the  lengths  of  the 
three  sides. 

30.  The  product  of  the  sum  and  the  difference  of  two  num- 
bers is  a;  the  quotient  of  the  sum  divided  by  the  difference 
is  b  ;  find  the  two  numbers. 

31.  A  rectangle  whose  dimensions  are  6  inches  and  10  inches 
is  to  be  doubled  in  area  by  increasing  the  length  and  width  by 
additions  proportional  to  the  present  dimensions.  Find  the 
necessary  addition  to  both  dimensions. 

32.  Solve  problem  31  if  the  area  is  to  be  made  four  times  as 
great. 

33.  The  dimensions  of  a  rectangular  piece  of  tin  are  in  the 
ratio  of  3  to  5.  Two-inch  squares  are  cut  from  the  corners 
and  the  sides  and  ends  are  turned  up  to  form  a  box.  What 
were  the  original  dimensions  of  the  rectangle  if  the  box  holds 
88  cubic  inches  ? 


XXIV.    GRAPHICAL  SOLUTION  OF  EQUATIONS 


570.  Graphical  Solution  of  Equations  Containing  One  Unknown 
Number.  In  §  369  a  system  of  two  linear  simultaneous  equa- 
tions was  solved  graphically.  A  linear  equation  in  one  un- 
known can  be  solved  graphically.     (Review  §§  356,  358,  359.) 


V 

1 

0/ 

i 

/ 

■ 

.V' 

.i 

X 

_ 

t 

5 

-i 

I 

- 

i 

0 

0 

I 

1 

. 

\ 

1 

-1 

/ 

/ 

-9 

1 

/ 

-a 

i 

/ 

A 

/ 

/ 

—6 

/ 

B 

/ 

V 

571.    Consider  the  equation  2  x  —  3  =  0.     For  definite  values 
of  x,  values  of  2  x  —  3  may  be  found  and  tabulated  as  follows  : 


Use  the  pairs  of  values  of  x  and  2  x  —  3  as  the 
coordinates  of  points  and  draw  the  line  BC  through 
these  points.     This  line  is  the  graph  of  2  x  —  3. 


1  This  chapter  may  be  omitted,  if  desired,  without 
interruptiDg  the  sequence  of  the  work. 
418 


X 

2z-3 

- 1 

-5 

0 

o 
—  f  J 

1 

- 1 

2 

1 

3 

3 

4 

5 

Graphical  Solution  of  Equations 


419 


This  graph  crosses  the  aj-axis  at  the  point  A  whose  coordi- 
nates are  (f,  0) ;  that  is,  at 
the  point  where  the  graph 
crosses  the  a>axis 

2x-3=0 
and  x  =  f . 

The  abscissa  of  the  point 
of  intersection  of  the  graph 
of  2  x  —  3  with  the  jr-axis 
is  the  root  of  2  x  —  3  =  0. 

572.    Consider  the  sec- 
ond degree  equation 
X2  _  x  _  6  =  0. 

We  shall  tabulate  values 
of  x  and  the  correspond- 
ing values  of  x2  —  x  —  6, 
and  use  these  numbers  as 
the  coordinates  of  points. 
This  will  give  us  the 
graph  of  x2  —  x  —  6. 

The  intersections  of  this  graph  with  the  cc-axis 
are  points  whose  abscissas  correspond  to  ordi- 
nates  0.  Since  the  ordinates  are  the  values  of 
x'2  —  x  —  6,  the  abscissas  whose  ordinates  are  0 
must  represent  the  values  of  x  for  which  x2  —  x 
—  6  =  0.  In  the  figure  these  values  of  x  are  3 
and  —  2.  These  numbers  are  the  roots  of  the 
equation  x2  —  x  —  6  =  0. 

The  result  may  be   checked  by  solving  this 
equation  by  one  of  the  algebraic  methods. 
_1  ±^/2E 
2 


1 

o 

7 

| 

I 

/ 

1 

i 

1 

/ 

\ 

/ 

\ 

\ 

•> 

1 

\ 

/ 

\ 

J 

f 

\ 

X' 

X 

-3 

-i 

- 

1 

0 

l 

1          3 

4 

\ 

/ 

' 

/ 

\ 

— ? 

/ 

-fl 

-| 

i 

/ 

-8 

/ 

V 

X 

x2  -  x  -  6 

4 

6 

3 

0 

2 

-4 

1 

-6 

0 

-  6 

-  1 

-4 

-  2 

0 

-3 

6 

Thus, 


3  and  -  2. 


The   graph  just  obtained  is  the  graph  of  the  expression 
x2  —  x  —  6,  or  of  the  equation  y  —  x2  —  x  —  6. 


420 


Graphical  Solution  of  Equations 


EXERCISE 

573.  Solve  graphically  as  in  §  572  the  following  equations  : 

1.  x2  —  x  —  8  =  0.  6.  2  x2  —  3  x  —  10  =  0. 

2.  x2  +  x-8  =  0.  7.  2  a2  +  2  a;  -10  =  0. 

3.  z2  +  <c-5  =  0.  8.  3a2-6a  +  2  =  0. 

4.  -a;2  -2a; +  7  =  0.  9.  2  a;2  +  5x  -  8  =  0. 

5.  2a;2  +  3a;-19  =  0.  10.  2a:2  -  5x  +  2  =  0. 

574.  The  accompanying  figure  shows  the  graphs  for  a  series 
of  expressions  of  the  form  x2  —  4  x  -f  &,  each  formed  from  the 

preceding  by  adding  a  posi- 
tive number  to  the  absolute 
term.  The  corresponding 
equations  are  as  follows  : 

1 .  x2  —  4  x  =  0. 

2.  x2  -  4  x  +  4  =  0. 

3.  x2-4x  +  8  =  0. 

The  addition  of  a  positive  num- 
ber to  the  absolute  term  increases 
by  the  number  added  the  value  of 
x2— 4x+k,  that  is,  of  the  ordinate 
corresponding  to  any  value  of  x. 
The  points  on  the  curve  are  all 
raised  equally  and  therefore  the 
graph  for  the  new  expression  is  of 
the  same  shape  as  that  of  the 
original  expression,  but  it  has  a 
different  position  relative  to  the 
x-axis.  In  fact,  the  first  two  terms 
of  the  expression  determine  the 
shape  of  the  graph,  while  the  abso- 
lute term  affects  its  position  verti- 
cally. 

In  the  figure  it  is  seen  that  in- 
creasing the  absolute  term  makes 
the  intersections  of  the  graph  with 


\Y               fl 

-      4                          t 

r-A                                           t-i' 

■V-                      -t-i 

t-8k-                            1     1 

It.                7    7 

1,1                  ±    t 

J    J                  L    t 

■    •       ,6       \                            1 

X  X       t  l 

^X  v     i  i^ 

YA     X     2     t-\ 

t-t   -F-^S     't 

X  V-           1  t 

4-4                t   t 

A   A                L   L 

L_S           ^ -J          —I 

XX           I   j 

4     V      -ft 

A      %      7      t 

*  ^      v^     t  x 

\0       1         2          3        4j           5 

-it             1 

A                 t 

=»  V          -t 

X       7 

=3   V        t 

\    / 

-i    ^y 

V 

Graphical  Solution  of  Equations 


421 


the  z-axis  approach  each  other  ;  therefore  this  change  makes  the  roots 
approach  each  other  in  value.  The  second  equation  has  equal  roots ;  its 
graph  is  tangent  to  the  ic-axis.  The  third  equation  has  imaginary  roots  ; 
its  graph  does  not  touch  the  x-axis. 


EXERCISE 

575.  1.  What  is  the  graphical  interpretation  of  the  absolute 
term  of  a  function  of  x  ?  of  the  absence  of  an  absolute  term  ? 

2.  How  will  the  graph  for  3  —  5  x  —  x2  differ  from  the  graph 
f  or  x*  +  5  x  -  3  ? 

3.  How  does  the  graph  for  x  —  3  differ  from  the  graph  for 
x-1? 

4.  Draw  the  graph  for  y  =  x  ;  y  =  x2. 

5.  Explain  how  the  graph  for  y  =  x2  can  be  used  for  finding 
squares,  or  square  roots,  of  numbers. 

6.  Draw  the  graph  for  y  =  x3. 

7.  Show  that  the  points  (1,  0)  and  (0,  —  1)  are  common  to 
the  graphs  of  y  =  x  —  1,  y  =  x2  —  1,  y  =  x?  —  1,  etc. 

576.  Consider  the  first  degree  simultaneous  equations 

2x  +  3y  =  6, 
Sx-2y  =  2. 

Tabulating  values  of  x  and  y  for  these  equations,  we  have 
the  following : 

2x  +  Sy  =  6.     (1)  Sx-2y  =  2.     (2) 


X 

y 

X 

y 

0 

2 

0 

- 1 

1 

4 

3 

1 

1 

2 

1 

2 

2 

3 

0 

3 

1 

-1 

! 

-1 

5 

-2 

¥ 

-2 

-4 

422 


Graphical  Solution  of  Equations 


Locating  the  points  corresponding  to  the  values  of  x  and  y  as  in  §§  356, 
359,  we  get  the  graphs  of  these  two  equations.  The  intersection  of  these 
lines,  P,  has  for  abscissa  approximately  1.4  and  for  ordinate  1.1.    These 

values,  x  =  1.4  and  y  =  1.1,  are 
common  to  both  lines,  that  is, 
to  both  equations. 

The  results  agree  closely 
with  those  obtained  by  solving 
the  system  by  one  of  the  usual 
methods. 

2  x  +  3  y  =  6. 

Sx-2y  =  2. 

6  x  +  9  y  =  18. 

6  a;  —  Ay  =  4. 
13  y=  14. 

y  =  H,  or  1.07+ 
x  =  Jf,  or  1.38+. 

Since  the  graph  of  a  linear 
equation  is  a  straight  line,  and 
a  straight  line  is  determined  by 
two  of  its  points,  it  will  not 
be  necessary  to  tabulate  as  many  values  as  were  shown  in  this  example. 
It  will  be  well,  however,  for  the  student  to  find  three  or  four  sets  of  values, 
as  the  additional  values  serve  as  a  check  on  the  first  two  computed. 


(1) 

V 

\ 

(Sj 

\ 

s 

v 

p 

/ 

N 

\ 

X 

/ 

s 

N 

X 

3 

_i 

| 

- 

i 

°/ 

r 

1 

i 

i 

s> 

s 

-i 

/ 

s 

— o 

-s 

y 

/ 

— 1 

/ 

Y\ 

EXERCISE 


577.    Solve  graphically  : 

1.  x  +  y  =  5, 

4  x  —  3  y  =  6. 

2.  x  +  2y  =  5, 
2x-y  =  0. 

3.  z  +  2/  =  0, 
3x-2/  =  2. 

4.  2x  —  y  =  5, 
3x  +  2y  =  -3. 

5.  x  =  2ij, 

x  +  2y  =  $. 


6.  5  a  +  4  2/  =  22, 
3a;  +  y  =  9. 

7.  7x-2?/  =  3l, 
4a>  +  3y=  —  3. 

8.  6x  +  lly  =  -2$, 
5y.-18*«-& 

9.  6#  +  22/  =  -  3, 
5#  —  32/  =  —  6. 

10.   4  a +  15?/ =  7, 
14  £  +  6?/  =  9. 


Graphical  Solution  of  Equations 


423 


578.    Consider  the  second  degree  system  : 

#2  -  V  =  4, 
x  +  2y  =  3. 

o-2-y  =  4.     (1)  x  +  2*/  =  3.     (2) 


x 

y 

X 

y 

0 

-4 

0 

a 

±1 

-3 

1 

1 

±2 

0 

3 

0 

±3 

5 

-3 

3 

Locating  the  points  tabulated  and  drawing  the  graphs  we  have  a 
curve  for  equation  (1)  and  a  straight  line  for  equation  (2).  The 
intersections  are  points  whose  coordinates  satisfy  both  equations  and 
therefore  give  the  roots  of  the  system. 

The  roots  are  approxi-      

mately 

86  =  2.1,    -2.7. 
y  =  .4f    2.8. 

If  solved  by  the  usual 
method,  we  find 

x  =  2.1+,    -2.6+. 
y  =  .45+,    2.8+. 

The  student  is  not  to 
understand  that  the  graph- 
ical method  of  solving  a 
system  of  simultaneous 
equations  is  to  replace 
the  algebraic  method.  The 
algebraic  method  is  gen- 
erally much  shorter  than 
the  graphical  method. 
However,  the  graphical 
method  of  representing 
equations  plays  a  very  im- 
portant part  in  higher 
mathematics  and  in  the  applications  of  mathematics  to  problems  of  physics 
and  engineering.     It  may  also  be  noted  that  the  algebraic  methods  do  not 


11             Y           1 

(1)1                                   (1) 

11 

L                       _4                            J 

^S                       -3                            l 

ZW(2)                r 

^^         2                             ' 

C     ^^^        u 

X~    - 1  ^  / 

X           ^t 

x'     J_                 S^,_   x 

-y     -r  -i      oi     /2     F^-J 

V       =1           I             * 

X          t 

U-      -2          — 4 

t        1 

3^=*  -t 

\     7 

^./ 

*" 

424 


Graphical  Solution  of  Equations 


always  furnish  the  solutions  of  simultaneous  quadratics  (§  551).  The 
graphical  method  can  generally  be  depended  upon  to  give  good  approxi- 
mations to  the  real  roots  in  such  cases. 


Solve  graphically : 


x2  +  y2  =  16, 
x2  —  y2  =  4. 


z2  +  y2 

=  16.     (1) 

X 

y 

0 

±4 

±1 

±  3.87+ 

±2 

±  3.46+ 

±3 

±  2.64 

±4 

0 

±5 

imag. 

**-i 

2  =  4.     (2) 

X 

y 

0 

imag. 

±  1 

imag. 

±2 

0 

±1 

4-    3 

±4 

±  3.46+ 

±5 

±  4.58+ 

The  intersections  give  the  roots  approximately  as  follows 

a;  =  8.1,  3.1-,    -3.1,   -3.1. 
y  =  2.5,    -2.5,   2.5,      -2.5. 


X" 

J 

7 

~F 

\m 

4 

(8)7 

i 

7 

X 

4- 

^v 

/ 

^5 

/ 

7    1 

\ 

r 

\ 

X' 

_  ± 

X 



5        — 

'i      H 

J        -j 

J2       - 

1          C 

i      l 

2       3 

4        5 

_J 

7 

-2 

r 

=8 

-a- -4 

1 

Graphical  Solution  of  Equations 


425 


The  algebraic  solution  gives 

x  =  Vl\  -VlO  or  3.16+,   3.16+,    -3.16+     -3.16+ 
y=±V6,    ±V6   or  2.44+,    -2.44+,   2.44+,    -2.44+. 

The  curve  for  equation  (2)  is  a  hyperbola.  The  curve  for  a  second 
degree  equation  in  two  unknown  numbers  is,  in  general,  a  circle,  a 
parabola,  an  ellipse,  or  a  hyperbola. 

579.  Imaginary  roots  cannot  be  found  by  this  method.  The 
presence  of  imaginary  roots  is  indicated  by  a  failure  of  the 
graphs  to  intersect.     Thus,  if  we  attempt  to  solve  the  system 


l 

Y 

[» 

(0 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

/ 

/ 

A" 

/ 

- 

I 

f 

1 

0 

1 

/ 

3 

'. 

\ 

5 

\ 

/ 

\ 

/ 

(2) 

-9 

f 

- 

*s4 

-5 

Y' 

x2  —  y  —  4,  (1),  x  —  y  —  5,  (2),  we  shall  find  that  the  graphs 
have  no  common  points.  The  graph  of  the  first  equation  is 
shown  in  §  578.  The  second  gives  the  line  (2)  as  shown  in  the 
figure.     The  algebraic  solution  of  this  system  gives 


_l± 


9  2 


426  Graphical  Solution  of  Equations 

EXERCISE 

580.    Solve  graphically  : 

1.   aj«  +  y*=:9,     (Circle.)  2.    a*  +  y*  =  9, 

x  -  ?/  =  0.  &  +  y  »  0. 

3.  a*  +  2  y2  _  2  a;  =  15,     (Ellipse.) 
sb  +  2  y  sb  1, 

4.  a2  -|-  #2  =  4, 

y  =  x  —  2  V2. 

Solve  example  4  also  algebraically. 

5.  a2  +  22/2_2a=15, 

aj8  _  2  2/2  =  _  7.     (Hyperbola.) 

6.  («-l)2-f-(2/_l)2  =  6,     (Circle.) 
x  -  y  =  0. 

7.    a2  +  a*/ +  ?/2  =9,  8.    «2  +  y  =  7, 

a*  +  ?/  =  9.  ^  +  2/2  =  11. 

Try  to  solve  example  8  algebraically. 


XXV.    THE  PROGRESSIONS 

ARITHMETICAL  PROGRESSION 

581.  Series.  A  succession  of  terms  formed  according  to 
some  definite  law  is  a  series. 

Thus,  I,  I,  \  •••  and  a,  a2,  a8  •••  are  series.  What  is  the  fourth  term  of 
each? 

582.  Arithmetical  Progression.  A  series  in  which  each  term 
after  the  first  is  found  by  adding  a  constant  quantity  to  the 
preceding  term  is  an  arithmetical  progression  (A.P.). 

583.  Common  Difference.  The  constant  number  added  is  the 
common  difference.  The  common  difference  is  found  by  sub- 
tracting any  term  from  the  term  immediately  following  it. 

Thus,  1,  3,  5,  7  •••,  and  12,  8,  4,  0,  —  4,  —  8  •••  are  arithmetical  pro- 
gressions. In  the  first,  2  is  the  common  difference  and  is  added  to  each 
term  to  form  the  next ;  in  the  second,  —  4  is  the  common  difference  and 
is  added  to  each  term  to  form  the  next. 

ORAL  EXERCISE 

584.  What  is  the  common  difference  in  each  of  the  following 
series  ? 

1.  7,  11,  15,  19,  — .  5.  a  —  x,  a,  a  -f-  x, 

2.  5,  8,  11,  14,  ....  6.  a  —  3  d,  a  —  d,  a  +  d  -. 

3.  i,  6J,  121  l.8|, ....  7.  a,  b,2b  -  a,  « , 

4.  a,  a  -f  d,  a  +  2  d,  a  -f  3  d,  — .  8.  1,  a,  2  a  —  1, 

9.  Form  the  next  two  terms  in  each  of  the  series  in  examples 
lto6. 

427 


428  The  Progressions 

585.  Last  Term.     In  the  arithmetical  progression  let 

a  represent  the  first  term, 
d  the  common  difference, 
n  the  number  of  terms, 
I  the  last  term,  and 
s  the  sum  of  terms. 

Then,  if  we  examine  the  series, 

First  term         Second  term  Third  term  Fourth  term         nth  term 

a,  (a  +  d),         (a  +  2e*),       (a  +  3d),      ...  a+(n-l)d, 

we  notice  that  in  any  term  the  coefficient  of  d  is  one  less  than 
the  number  of  the  term  in  the  series. 

Hence  in  a  series  of  n  terms,  the  nth  term  being  the  last, 

7=a+(n-l)d.  (A) 

1.  Find  the  7th  term  of  the  series  4,  2,  0,  -  2,  .... 

Solution.     In  this  series  a  =  4,  d  =  —  2,  n  =  7. 
.-.  Z  =  4  +  6-(-2)  =  -8. 

2.  Find  the  first  4  terms  and  the  last  term,  when  a  =  2, 
d  =  |,  w  =  8. 

Solution.     The  first  four  terms  are  2,  2f ,  2f,  2$, 

Z  =  2  +  7  •  f  =  4. 

586.  Sum  of  the  Terms.     To  find  the  sum  of  a  number  of  terms 
in  arithmetical  progression : 

Write  the  sum  of  the  series  in  the  usual  order,  (1),  and  in  reverse 
order,  (2),  and  add  the  two  equal  series. 

(1)  S  =  a         +(a  +  d)  +  (a+2e*)  +  (a  +  3d)+  •••  +(J-d)  +1 

(2)  S=l  +(l-d)  +(Z-  2d)  +(l-3d)  +  .»  +(g  +  d)+a 

2  S  =  (a  +  l)  +  (a  +  I)  +(a  +  I)    +(a  +  l)     +  ...  +(a  +  J)  +  (a  +  0 
=  n(a  +  0.     (Why?) 


Arithmetical  Progression  429 

Substituting  the  value  I  =  a  +  (n  —  l)d  from  (A),  we  can  get  a  formula 
for  the  sum  in  terms  of  a,  n,  and  d. 

S  =  5[8a  +  (n-l)d].  (C) 

1.  Find  the  sum  of  the  terms  of  the  series,  2,  5,  8,  11,  — ,  to 
12  terms. 

Solution.     I  =  2  +  11  •  3  =  35. 

fl  =  ?  (a  +  Z)  =  1?  (2  +  35)  =  222. 

A  A 

2.  Find  the  sum  of  the  series  4,  2,  0,  —  2  — ,  to  20  terms. 

Solution.        In  the  series  a  =  4,  d  =  —  2,  and  w  =  20. 
Hence,  using  formula  (C) ,  S  =  22° ■  [2  •  4  +  (20  -  1)  (  -  2)] 
=  10(8  -  38)  =  -  300. 

3.  The  first  term  of  a  series  is  5,  the  last  term  is  161,  and 
the  sum  of  the  series  is  3320.  Find  the  number  of  terms  and 
the  common  difference. 

Solution.     Using  formula  (5),  3320  =  -  (5  +  161)  =  83 n. 

A 

.-.  n  =  40. 
Using  formula  (A),  161  =  5  +  39  d. 

.\d  =  4. 


587.  An  arithmetical  progression  can  be  completely  deter- 
mined if  any  two  of  its  terms  are  known. 

The  6th  and  15th  terms  of  an  A.  P.  are  14  and  32,  respec- 
tively.    Find  the  20th  term. 

Solution.  a  +  5  d  =  14, 

and  a  +  14  d  =  32. 
.  •.  d  =  2  and  a  =  4. 
Hence  the  20th  term  =  4  +  19  .  2  =  42. 

588.  Arithmetical  Mean.  When  several  quantities  are  in 
A.  P.  the  terms  between  the  first  and  last  terms  are  the 
arithmetical  means  between  them. 


430  The  Progressions 

The  arithmetical  mean  between  two  numbers  is  equal  to  one  half 
their  sum. 

Proof.     Let  a  and  b  be  two  numbers  and  A  their  arith- 
metical mean.     Since  a,  A,  b  are  in  arithmetical  progression, 
b  -  A  =  A  -  a,     (Why  ?) 

otA  =  <L±*. 
2 

Any  number  of  arithmetical  means  may  be  inserted  between 
two  numbers  by  means  of  formula  (A). 

Insert  10  arithmetical  means  between  10  and  72. 

Solution.     In  this  case  a  =  10,  I  =  72,  n  =  12.     (Why  ?) 
Substituting  in  (A),       72  -  10  +  11  d. 
•••<*  =  If  =  5!V 
Therefore  the  series  is  10,  15T^-,  21T3T,  etc. 

589.  If  any  three  of  the  five  numbers  a,  d,  n,  I,  and  S  are 
known,  it  is  possible  to  find  the  other  two  from  one  or  both 
of  the  formulas 

l=a  +  (n-l)d.  {A) 

S=?(a+7).  (B) 

It  will  be  noted  that  four  of  the  five  numbers  involved  in  an 
A.  P.  are  found  in  each  formula. 

In  (A),  we  have  a,  n,  d,  I. 

In  (B),  we  have  a,  n,  I,  S. 

In  order  to  use  the  formulas  in  the  solution  of  problems  we 
need  to  know  three  of  the  five  numbers.  If  the  three  given 
numbers,  and  the  one  required,  are  all  found  in  one  formula, 
the  problem  may  be  solved  from  that  formula  alone. 

1.    Given  a  =  5,  n  =  7,  I  =  15,  find  d. 

Solution,    a,  Z,  n  and  d  are  all  in  (A).     From  it  we  may  write 
15  =  5  +  6  •  d. 


Arithmetical  Progression  431 

2.  From  the  data  in  example  1,  find  S. 

Solution.     Substituting  their  values  for  a,  »,  and  I  in  (2?)  we  have 
S  =  |(6  4- 15)  =70. 

3.  Write  the  first  four  terms  of  the  series  from  the  data  in 
example  1. 

Solution.  First  proceed  as  in  the  solution  of  example  1  for  d  =  f . 
Then  we  have,  for  the  series, 

6,  6f,8§,  *>... 

590.  If  the  three  numbers  given  and  the  number  required 
are  not  all  found  in  either  (A)  or  (B)  alone,  these  formulas  may 
be  treated  as  a  pair  of  simultaneous  equations  after  the  proper 
substitutions  have  been  made. 

Given  d  =  2,  I  =  20,  S  =  108  ;  find  a  and  n  and  the  series. 
Solution.    From  (4),  20  =  a  +  (n  -  1  )2,  or  22  =  a  +  2  n. 

From  (JB) ,  108  =  -  (a  +  20)  or  216  =  an  +  20  n. 

2i 

This  gives  us  the  simultaneous  system  a  +  2  n  =  22,  aw  +  20  n  =  216. 

Solving,  a  =  22  -  2  w. 

216=  (22-2w)n+20n. 

2n2-42n  +  216  =0. 

n2  _  21  n  +  108  =  0. 

(W_9)(w-12)=0. 

.-.  n  =  9  or  12. 

When  n  =  9,  a  =  4. 

When  n  =  12,  a  =  -  2. 

The  series  is  either  4,  6,  8,  ••-,  20,  or  -  2,  0,  2,  4,  •••,  20. 

EXERCISE 

591.  1.  Show  that  the  three  numbers,  x  —  y,x,x  +  y  form 
an  A.  P.     Similarly  f  or  x  —  3  y,  x  —  y,  x  +  y,  x  +  3  y. 

2.  a  =  —  3,  d  =  2,  n  =  8  ;  find  Z  and  #. 

3.  a  =  3,  d  =  3,  Z  =  15 ;  find  n  and  #. 

4.  a  =  4,  7i  =  12,  I  =  26 ;  find  d  and  A 


432  The  Progressions 

5.  d  =  I-,  n  =  3,  I  =  2  ;  find  a  and  & 

6.  a  =  15,d  =  -i,  #  =  1371;  find  n  and  Z. 

7.  a  =  4,  »  =  15,  £  =  270 ;  find  d  and  Z. 

8.  d  =  2,  n  =  15,  S  =  270 ;  find  a  and  Z. 

9.  a  =  10,1  =  37,  5  =  235 ;  find  d  and  n. 

10.  d=-2,  J  =  -24,  #  =  -144;  find  a  and  n. 

11.  n  =  13,  I  =  41,  #  =  299 ;  find  a  and  d. 

12.  Find  4  arithmetical  means  between  5  and  18. 

13.  Insert  6  arithmetical  means  between  —  5  and  13. 

14.  The  snm  of  three  terms  of  an  A.  P.  is  45 ;  the  sum  of 
the  squares  of  the  terms  is  773.     Find  the  series. 

Note.    Use  x  —  y,  x,  and  x  +  y  to  represent  the  series. 

15.  How  many  times  does  the  clock  strike  in  12  hours  ? 

16.  Find  the  sum  of  the  first  20  odd  numbers. 

17.  Find  the  sum  of  the  first  20  even  numbers. 

18.  Show  that  the  sum  of  the  first  n  natural  numbers  is 
n(n  +  1) 

2 

19.  If  you  save  1  ^  today,  2  ^  tomorrow,  3  ^  the  next  day, 
and  so  on,  how  many  days  will  elapse  before  the  total  savings 
amount  to  $  10  ? 

20.  The  fifteenth  and  twenty-eighth  terms  of  an  A.  P.  are 
respectively  12  and  19.     Find  the  first  and  the  fiftieth  terms. 

21.  Insert  four  arithmetical  means  between  9  and  11. 

22.  The  sum  of  the  first  8  terms  of  an  A.  P.  is  64  and  the 
sum  of  the  first  18  terms  is  324.     Find  the  series. 

23.  The  sum  of  the  first  7  terms  of  an  A.  P.  is  7  and  the 
sum  of  the  next  8  terms  is  68.     Find  the  series. 

24.  Between  6  and  10,  there  are  12  numbers  so  that  the 
whole  series  of  14  numbers  forms  an  A.  P.  What  is  the  sum 
of  the  series  ? 


Arithmetical  Progression  433 

25.  The  sum  of  the  third  and  fifth  terms  of  an  A.  P  is  32? 
and  the  sum  of  the  fourth  and  tenth  terms  is  50.  Find  the 
first  term,  and  the  sum  of  the  first  20  terms. 

26.  Twenty  potatoes  are  laid  out  in  a  straight  line  one  yard 
apart.  How  far  must  a  boy  run  to  pick  them  up  and  bring 
them,  one  at  a  time,  to  a  basket  placed  in  the  line  and  one  yard 
from  the  first  potato  ? 

27.  A  freely  falling  body  falls  |  g  feet  the  first  second,  f  g 
feet  the  second  second,  %g  feet  the  third  second,  and  so  on. 
How  far  will  it  fall  in  t  seconds  ? 

28.  If  g  =  32.16  feet,  through  what  distance  does  a  body  fall 
if  it  reaches  the  ground  in  6  seconds  ?  How  far  does  it  fall  in 
the  6th  second  ? 

29.  If  a  ball  is  dropped  from  the  top  of  Washington  Monu- 
ment, 550  ft.  high,  how  long  does  it  take  to  reach  the  ground  ? 

30.  How  long  does  it  take  the  ball  in  problem  29  to  get 
halfway  to  the  ground  ? 

31.  In  Italy  24-hour  clocks  are  used.  How  many  strokes 
does  such  a  clock  strike  in  a  day  ? 

32.  In  an  A.  P.  of  ten  terms  the  product  of  the  first  and  last 
terms  is  70  and  the  sum  of  all  is  95.     Find  the  series. 

33.  How  many  numbers  of  2  figures  are  divisible  by  3  ? 
(a  =  12,1  =  99,  d  =  ?) 

34.  Find  the  sum  of  all  numbers  of  two  figures  each  that 
are  divisible  by  3  ? 

35.  What  is  the  sum  of  the  first  50  multiples  of  7  ? 

36.  The  sum  of  n  terms  of  the  series  2,  5,  8,  •••,  is  950. 
Find  n. 

37.  The  sum  of  n  terms  of  the  series  87,  85,  83,  •••,  is  the 
same  as  the  sum  of  n  terms  of  3,  5,  7,  •••.     Find  n. 


434  The  Progressions 

GEOMETRICAL   PROGRESSION 

592.  A  geometrical  progression  (G.  P.)  is  a  series  in  which 
each  term  after  the  first  is  derived  by  multiplying  the  preced- 
ing term  by  a  constant  multiplier  called  the  ratio. 

Thus,  3,  6,  12,  24,  ••-,  and  36,  -  6,  1,  —  £,  •••,  are  geometrical  pro- 
gressions.   The  ratios  are  respectively  2  and  —  \. 

593.  Ratio.  The  ratio  (r)  is  found  by  dividing  any  term 
by  the  term  immediately  preceding  it. 

ORAL  EXERCISE 

594.  What  is  the  ratio  in  each  of  the  following  series? 
1.   2,  6,  18,  54,  ....  5.   1,  V2,  2,  .... 

2-    12,6,3,|,....  6.    «,6,  ^,  .., 

3.  5,-10,20,-40.  i        * 

i  .       _L,  J.,    _L,      ". 

4.  a,  ar,  ar2,  ar3.  8.    1,  i,  —  1,  •••. 

9.  Form  the  next  two  terms  in  each  of  the  series  in  ex- 
amples 1  to  8. 

595.  Last  Term.  If  a  is  the  first  term,  I  the  last  term,  r  the 
ratio,  and  n  the  number  of  terms,  we  have  the  following  from 
the  definitions  : 

1st  term  2d  term  8d  term  4th  term  5th  term  nth.  term 

a  ar  ar2  ar3  ar4      ••♦       arn_1 

By  examining  this  series  we  notice  that  the  exponent  of  r  is 
always  one  less  than  the  number  of  the  term  in  the  series. 
Hence,  in  a  series  of  n  terms,  the  nth  term  being  the  last, 

/  =  ar"-*.  (A) 

Thus,  the  8th  term  of  3,  f ,  f,  •••,  is  3  •  Q)7  =  Tf*,  and  the  last  term  of 
1,  5,  25  to  10  terms  is  I  =  1  •  59  =  59. 


Geometrical  Progression  435 

596.   Sum  of  the  Terms.     To  find  the  sum  of  the  terms  in  a 
geometrical  progression : 

Write  the  sum  of  the  series, 

S  =  a  +  ar  +  ar2  +  ar3  +  •••  +  ar""1  (1) 

(1)  x  r,  rS  =         ar  +  ar2  +  ar8  +  •  •  •  +  ar"-1  +  ar"         (2) 

(l)-(2)  S-rti=a  -ar" 

,.S  =  ^=^or^^.  (J) 

1  —  r  r  —  1 

Since  or"-1  =  Z  or  arn  =  rl,  the  formula  may  be  written 

a  formula  that  is  sometimes  useful. 

Find  the  sum  of  6,  3,  1-J,  •••,  to  10  terms. 
Solution,     a  =  6,  r  =  |,  n  =  10. 

.-.from  (2?),  we  have  £  =  G  ~  6(i)10=  g^ff  =  lif  5f. 

*     y  1  -  1  256  *5* 


597.  Geometrical  Mean.  If  several  quantities  are  in  G.  P., 
the  terms  between  the  first  and  last  terms  are  the  geometrical 
means  between  them. 

The  geometrical  mean  between  two  numbers  is  the  square  root  of 
their  product. 

Proof.     Let  G  be  a  geometrical  mean  between  a  and  b. 

—  =  — ,  for  each  fraction  equals  the  ratio  of  the  series. 
a      G 

.-.  G2  =  ab  and  Q  =  ±  Vo&. 
Find  a  geometrical  mean  between  V8  and  V2. 


Solution.     G  =  ±v  V8  •  V2  =±  2. 

The  student  should  notice  that  the  geometrical  mean  and 
the  mean  proportional  are  the  same.     See  §  326. 


436  The  Progressions 

We  may  also  insert  several  geometrical  means  between  two 
given  numbers. 

Insert  4  geometrical  means  between  16  and  ^p. 

Solution.  Here  a  =  16,  n  =  6,  (Why  ?)  I  =  *$* 

From  (A),  ^  =  16^. 

r*  =  W- 
r  =  f. 

Let  the  student  complete  the  solution. 

598.  Application  of  the  Formulas.  The  formulas  in  geometri- 
cal progression  to  be  remembered  are  : 

l  =  arn~K  (A) 

s  =  a^a0Ta-ar^t 
r—  1  1  —  r 

S  =  ^or^.  (C) 

r- 1        1  - r 

In  (4),  we  have  a,  w,  r,  Z. 

In  (B),  we  have  a,  n,  r,  JS. 

In  (C),  we  have  a,  r,  Z?  S. 

599.  The  suggestions  of  §  589  hold  here  except  that  we  may 
not  be  able  to  solve  for  n  or  r.  In  most  cases  the  use  of  n  as 
an  unknown  introduces  equations  of  a  type  wholly  new  to  the 
pupil,  that  is,  with  the  unknown  number  an  exponent.  When 
r  is  unknown  it  may  become  necessary  to  extract  roots  higher 
than  the  second  or  third.  Both  these  problems  can  be  solved 
by  inspection  in  some  simple  cases ;  for  example,  2n  =  8, 
.-.  n  =  3 ;  and  r5  =  243,  .-.  r  =  3. 

Logarithms  may  also  be  used  in  such  solutions. 

1.   Given  a  =  1,  I  =  2,  n  =  4 ;  find  r. 

r,  a,  Z,  and  n  are  all  in  formula  (A). 
Hence  we  may  write  2  =  1  •  r3. 
r3  =  2. 

The  series  is  1,  y/2,  y/4,  2. 


Infinite  Geometrical  Series  —  Repeating  Decimals  437 

2.   From  the  data  in  example  1  find  S. 

Both  formulas  for  S  involve  r.     Using  the  result  obtained 
in  example  1,  we  may  write  from  (c), 

»=2v/2-l 
W-l  ■ 
The  result  may  be  left  in  this  form. 

EXERCISE 

600.  1.    Find  the  6th  term  of  1,  2,  4,  8,  .... 

2.  Find  the  sum  of  1  +  2  +  4  •••  to  6  terms. 

3.  Insert  3  geometrical  means  between  5  and  8. 

4.  Find  the  difference  between  the  arithmetical  mean  and 
geometrical  mean  of  1  and  2. 

5.  The  fourth  term  of  a  G.  P.  is  54  and  the  fifth  term  is 
486.     Find  a  and  r. 

6.  Find  the  sum  of  the  first  five  terms,  when  a  =  1,  r  =  f . 

7.  Find  a  fraction  whose  value  is 

l  +  aj  +  a£-fa?  +  .-.a;15. 

8.  Find  the  sum  of  the  first  5  terms  of  1,  -J,  J,  •••. 

9.  Find  the  geometrical  mean  between  14  and  686.     Be- 
tween 38  and  123  to  two  decimal  places. 

10.  If  Z=  128,  r  =  2,  n  =  7,  find  a  and  S. 

11.  If  a  =  9,  I  =  2304,  r  =  2,  find  S  and  n. 

12.  If  a  =  2,  Z  =  1458,  S  =  2186,  find  r  and  w. 

INFINITE   GEOMETRICAL   SERIES  —  REPEATING 
DECIMALS 

601.  The  student  will  recognize  the  identity 

£  =  .3333... 

This  means  that  the  repeating  decimal  .333  •••  approaches  in 
value  the  fraction  £. 


438  The  Progressions 

It  is  evident  also  that  the  repeating  decimal  equals 

.3  -h  .03  +  .003  +  -. 

This  is  a  geometrical  series  with  a  =  .3,  r  =  .1  and  n  indef- 
initely large. 

Using  formula  (B),  we  have 

q  _  a  —  arn  _     a  arn 

1  —  r       1  —  r     1  —  r 

.3         .3  x  .ln 


l-.l       l-.l 

_1      .3  x  .1" 
3  .9 

O  1 

The  second  term,  —  x.ln=-x.ln,  becomes  smaller  as  n  becomes 
larger.  Thus,  when  w  =  6,  -x  .ln  =  -  of  .000001.  When  n  becomes 
infinitely  large  the  term  becomes  so  small  that  it  may  be  neglected,  and 
we  have  the  sum  of  .3  +  .03  +  .003  +  •••  indefinitely  =  -. 

o 


602.  Formula  for  an  Infinite  Geometrical  Series.  The  result 
found  in  the  last  article  may  be  generalized  in  the  following 
statement : 

When  r  is  less  than  unity  and  the  number  of  terms  is  infinite, 

■n  &      a  —  arn         a  arn 

Proof.  S  = 


1  —  r       1  —  r      1  —  r 

When  r  <  1  and  n  is  infinitely  large,  rn  is  smaller  than  any 

assignable    number   and    therefore   the    term may  be 

neglected.     This  leaves 

S=     a 


1-r 


Infinite  Geometrical  Series  —Repeating  Decimals   439 

1.   Find  the  value  of  l-}-|  -f  \  ♦••  to  an  infinite  number  of 

terms.  i 

a  =  1,  r  =  £,  £ 


1-1 
2.    Find  the  value  of  3.2727  .... 

Solution.     Note  that  3  is  not  part  of  the  infinite  geometrical  series 
that  follows  it.     First  find  the  value  of  .2727  •  ••  =  .27  +  .0027  -f  ••••    Here 

a  =  27,  r  =  .01. 
B_     .27    _.27_3| 
1  -  .01      .99     11 ' 
.-.  3.2727.-.  =3^. 

EXERCISE 
603.    Find  I  and  S : 

1.  When  a  =  2,  r  =  2,  n  =  7. 

2.  When  a  =  5,  r  =  4,  n  =  9. 

3.  When  a  =  6,  r  =  f ,  n  =  6. 

4.  a  =  40,  r  =  f ,  n  =  oo  ;  find  S. 

(ao  is  the  symbol  for  an  infinitely  large  number.) 

5.  a  =  9,  r  =  |,  ?i  =  oo  ;  find  £. 

6.  If  the  first  term  of  a  geometrical  series  is  a,  and  the 
second  term  is  b,  what  is  the  ratio  ? 

7.  What  is  the  sum  of  the  first  four  terms  of  the  series  in 
example  6  ? 

8.  If  a  >  b  and  n  is  infinite,  show  that  the  value  of  S  in 

example  6  is 


a  —  b 


9.   Find  the  sum  of  the  infinite  series  -7  +  -, r— 

m  +  1      (m  -f  l)2 

when  m  >  0. 


(m  +  1)3 

10.  What  is  the  significance  of  making  m  >  0  in  example  9  ? 
Find  the  series  and  the  answer  in  9  when  m  =  2. 

11.  What  common  fraction  reduces  to  the  repeating  decimal 
.777-.? 

12.  Find  the  fractional  form  for  3.25757  •••. 


440  The  Progressions 

Some  of  the  problems  that  follow  will  require  the  use  of  the 
formulas  of  A.  P.,  and  some  will  be  in  G.  P. 

13.  How  many  numbers  of  two  figures  each  are  exactly 
divisible  by  7  ? 

14.  How  many  numbers  of  three  figures  each  are  multiples 
of  7? 

15.  How  many  numbers  under  1000  are  powers  of  2  ? 

16.  What  is  the  sum  of  all  the  three  figure  numbers  that 
are  multiples  of  5  ? 

17.  In  an  A.  P.,  S  =  20  +  13  V2,  n  =  10,  I  =  2.6  V2  ;  find  a 
and  d. 

18.  What  kind  of  series  is  -  2,  V2  -  1,  2  V2,  ...  ?     Write 
the  next  two  terms. 

19.  Given  a  +  V2,  2a  +  2,  3a  +  2V2,  4a+4-...      Show 
that  the  sum  of  this  series  to  10  terms  is  55  a  +  31(2  -f  V2). 

20.  Indicate  the  sum  of  n  terms  of  the  series  in  example  19. 

21.  The  first  term  of  an  infinite  geometrical  series  is  3  and 
the  second  term  is  2.     Find  the  sum. 

22.  Find  the  10th  and  15th  terms  of  2LzJt,  a-2    a-S    ^ 

a  a  a 

(Williams  College.) 

23.  Find  the  sum  of  10  terms  of  6,  —  4,  f  ...  and  the  sum  of 
12  terms  of  -  5,  -  1,  3  ....  (Williams  College.) 

24.  Determine  whether  3§,  4|,  6f  •••  are  in  A.  P.  or  G.  P. 
and  find  the  sum  of  the  first  6  terms  by  the  general  formula. 

25.  Find  the  geometrical  mean  between  6  -f  V2  and  6  —  V2. 
Find  the  arithmetical  mean  between  the 
same  numbers. 

26.    The  side  of  an  equilateral  triangle  is 
10  inches.     The  midpoints  of  its  sides  are 
joined,  forming  another  equilateral  triangle, 
and  this  process  is   repeated  indefinitely. 
Find  the  sum  of  all  the  lines. 


Infinite  Geometrical  Series  —  Repeating  Decimals  441 


27.  Lines  are  drawn  joining  the  middle  points  of  the  sides 
of  a  square,  thus  forming  a  second  square,  and  the  middle 
points  of  the  sides  of  this  square  are  joined. 
If  this  process  is  repeated  indefinitely,  find 
the  sum  of  all  the  lines,  if  a  side  of  the  origi- 
nal square  is  6  inches. 

28.  What   number   added   to  each  of   the 
numbers  1,  8,  22  will  make  a  G.  P.  ? 

29.  What  distance  will  an  elastic  ball  travel  before  coming 
to  rest  if  it  falls  20  feet  and  rebounds  each  time  f  of  the  dis- 
tance of  its  last  fall  indefinitely,  that  is,  until  it  comes  to  rest  ? 

30.  The  difference  between  two  numbers  is  48.  Their 
arithmetical  mean  exceeds  their  geometrical  mean  by  18.  Find 
the  numbers.  (Yale.) 

31.  Find  the  sum  of  n  terms  of 


<?-y)+{%   x2 


■■J  W    <ey 


(Yale.) 


3&3 


XXVI,    THE  BINOMIAL  FORMULA 

604.  By  means  of  the  binomial  formula  it  is  possible  to  raise 
a  binomial  to  any  required  power  without  actually  performing 
the  multiplications. 

The  following  is  the  binomial  formula : 

(a  +  b)n=  a»+  nan-*b+  n(n  ~  1)fln-2&2_L  n(n  ~  1)(72  ~  2)  an- 
v         J  12  123 

<n-l){n-2)(n-Z)  ^ 

T  1234  r     * 

The  proof  of  this  formula  will  be  assumed,  but  the  student 
should  note  that  the  following  results  obtained  by  using  the 
formula  agree  with  the  results  obtained  by  actual  multiplication. 

For  n  =  2, 

(a  +  b)2  =  a2  +  2  aV-Vb  +  2  '  j?  ~  *)  a**b*  =  a2  +  2  ab  +  62. 

1  •  ^ 

For  yi  =  3, 

a  +  &)3  =  a3  +  3  a*-1*  +  3  ' (3  ~  ^  a3~262  +  3(3~1)  (3  ~2)  a3~363 

1  •  Z  x  •  Z  •  o 

=  az  +  3a*b  +  3ab*  +  b\ 
Let  the  student  verify  the  formula  for  (a  +  6)4. 

605.  By  observing  the  formula,  we  note  the  following  points 
which  may  be  used  as  a  rule  : 

In  the  expansion  of  (a  +  b)n  : 

1.  The  number  of  terms  is  n  +  1. 

2.  The  first  term  is  an. 

3.  The  result  is  in  descending  powers  of  a  and  ascending  powers  of  b, 
b  appearing  first  in  the  second  term. 

4.  Each  coefficient  is  found  from  the  preceding  term  by  multiplying 
the  coefficient  of  that  term  by  the  exponent  of  a  and  dividing  the  result 
by  the  exponent  of  b  plus  1. 

442 


The  Binomial  Formula  443 

1.    In  the  expansion  of  a  -f-  b  to  a  certain  power  one  of  the 
terms  is  792  abb7.     What  is  the  next  term  ? 

Solution.     Applying  part  4  of  the  above  rule,  we  have  for  the  co«ffi- 

cient792x6  =  495. 
8 

Therefore  the  next  term  is  495  a468. 


2.  Expand  (a3  +  62)6  by  the  binomial  formula. 

Solution.     (a8+62)6  =  (a3)6  +  6(a3)5(&2)  +  15(a3)4(62)2  +  20(a3)3(62)3 
+  15(a3)2(62)4  +  6(a»)  (6*)*  +  (&2)« 
=  a18  +  6  a1562  +  15  a12*)4  +  20  a966  +  15  a6&8 
+  6  a3610  +  612. 

3.  Expand  (2  a;*  -  y~3)b. 

Solution.     Write  this,  [(2se*)  +  (  —  y-3)]5.     Here  a  is  2  a;2  and  6  is 
—  2T3.     Then 

[  (2  a£)  +  ( -  2T3)]5=  (2  x*)5  +  5(2  a£)*(  -  *r3)  +  10(2  z*)8(  -  jr3)2 

+  10(2  «*)«(-  *T3)3  +  5(2  **) (-  *T3)4  +  (-  *r3)5 

=  32  cc*  -80  z2?r3+80x  Vc-40  xyr»+ 10  xV12-*T16- 

EXERCISE 
606.    1.    In  expanding  a  binomial  by  the  formula : 

(1)  How  does  the  exponent  of  the  first  term  compare  with 
the  power  of  the  binomial  ? 

(2)  What  is  the  exponent  of  the  first  term  of  the  binomial 
in  each  term  of  the  expansion  after  the  first  ? 

(3)  In  what  term  of  the  expansion  does  the  second  term  of 
the  binomial  first  appear  ? 

(4)  How  does  its  exponent  change  from  term  to  term  ? 

(5)  What  is  the  coefficient  of  the  first  term  of  the  expansion  ? 
of  the  second  term  ? 

(6)  How  is  the  coefficient  of  each  term  after  the  second 
formed  ? 

(7)  How  many  terms  are  there  ? 


444  The  Binomial  Formula 

Expand  by  the  binomial  formula  and  simplify  the  terms : 

2.  (a -b)4.  13.  (|  +  2z*)5. 

3.  (a  +  2  b)5.  14.  (x  +  ar1)8. 

4.  (a»-3  6)«.  15.  fa_b\\ 

5.  (2a-|2/2)5.  \6      a/ 

7-    ^-2^  17.    (l+V=I)l 

8.    (3a;  +  2/)8.  lg     (1  _  ,.)8> 

9-  (5-2^)6.  19.  (l+V^3)3. 

10.  (tf-y)1.  20.  (3+V^5)7. 

11.  (f-fa)7-  21.  (a  -\-biy-  (a  -bi)*. 

12.  (±-3yy.  22.  (1  +  Va)7  -  (1  -  Vz)7. 

23.  What  are  the  signs  of  the  terms  in  the  expansion  of 
(a -b)7?  of  (-a  -{-b)7?  of  (-a +6)6?  of  {-a -by?  of 
(_a_6)6? 

24.  Expand  (2-3i)5. 

25.  Find  the  first  4  terms  and  the  last  term  of  (a^  —  6*)20. 

26.  Find  the  first  fonr  terms  and  the  last  term  of  (a  +  b)m. 

27.  How  many  terms  are  there  in  the  expansion  of  (a  -f  b)b? 
of  (a  +  b)6?  of  (a  +  b)n? 

28.  What  are  the  exponents  of  a  and  b  in  the  fourth  term 
of  (a  +  6)5?  in  the  fifth  term  of  (a  +  b)b?  in  the  fourth  term 
of  (a  +  b)6  ?  in  the  fifth  term  of  (a  +  6)6  ? 

29.  What  is  the  sum  of  the  exponents  of  a  and  b  in  each 
term  of  the  expansion  of  (a  +  b)5  ?  of  (a  +  &)6  ?  of  (a  +  6)n  ? 

607.  Binomial  Coefficients  and  Exponents.  The  coefficients  in 
the  expansion  of  a  -f  6  to  any  power,  and  the  exponents  of  a 
and  b  in  this  expansion,  are  called  respectively  the  binomial 


The  Binomial  Formula 


445 


coefficients  and  the  binomial  exponents  of  the  expansion.  These 
terms  are  used  to  distinguish  them  from  the  reduced  results  in 
cases  when  a,  for  example,  is  represented  by  2  x*  and  b  by  a 
similar  expression.  The  expansions  should  always  be  made 
first  in  binomial  coefficients  and  exponents.     (Why  ?) 

An  interesting  relation  among  the  binomial  coefficients  of 
successive  powers  of  a  binomial  is  shown  in  the  following 
scheme  known  as  Pascal's  Triangle  : 

The  numbers  in  the  first  line  are 
the  coefficients  of  (a  4-  b)  ;  in  the 
second  line  of  (a  -f  b)2 ;  in  the  third 
line  of  (a  4-  b)3,  etc. 

Any  number  in  this  scheme  equals  the 
number  directly  over  it  plus  the  number 
at  the  left  of  the  one  over  it.     The  coeffi- 
cients of  the  5th  power  of  a  +  b  are  found  from  the  coefficients  of  the 
fourth  power  as  follows  : 

1+4  =  5;  write  5  under  4  ;  4  +  6  =  10 ;  write  10  under  6  ;  etc. 

Let  the  student  write  the  coefficients  for  n  —  1  and  n  —  8. 


w  =  l 

1 

1 

n  =  2 

1 

2 

1 

w  =  3 

1 

3 

3 

1 

n  =4 

1 

4 

6 

4 

1 

n  =  5 

1 

5 

10 

10 

5 

1 

w  =  6 

1 

6 

15 

20 

15 

6 

M 

608.  Any  Required  Term.  Writing  the  binomial  formula 
and  numbering  the  terms,  we  may  make  a  formula  for  any 
term. 

1st  2d  3d  4th 

(a+6)»=a»+»a»-'6  +  2^1a-W  +  ^W7^("~2)  a-W+.~ 

l.  •  Z  1  •  Z  •  o 

6th 

»(n-l)(»-2)(n-3)(n-4) 
+  1-2.3.4.5  a     °  + 

10th 

n(n-l)(n-2)-.(n-8)  aB_969      ... 
I.2.3-9  T      * 

A  simple  way  to  find  any  required  term  in  the  expansion  of 
(a  +  b)n  is  to  start  with  the  exponent  of  6. 

What  is  the  exponent  of  b  in  the  2d  term?  in  the  3d 
term?   in   the   4th   term?      How  does  it   compare  with  the 


446  The  Binomial  Formula 

number  of  the  term  ?  In  the  rth  term  it  is  r  —  1.  In  a  similar 
way  the  exponent  of  a  is  n  minus  the  exponent  of  b;  that  is, 
n  —  (r  —  1)  or  n  —  r  +  1.  By  further  comparison  of  the  coeffi- 
cient of  any  term  with  the  number  of  the  term  it  will  be  seen 
that  the  rth  term  is 

1.2.3..(r-l)  r    ' 

1.  Find  the  7th  term  of  (1  a2  —  2  b%)12. 

First  write  the  binomial  in  the  form  [(Ja2)  +  (—  2  6*)]12. 
Applying  the  formula  for  the  rth  term  when  r  =  7  and  n  =  12, 
3    4 
we  have  ^'^'/^l^a a2)6(-  2 &*)6  =  924  oK&». 

2.  Find  the  middle  term  of  (a?  -  Vy)14. 

How  many  terms  are  there  in  the  expansion  ?     What  is  the 
number  of  the  middle  term  ? 

2    3 
l-g^V/fl-V   ^7(-VF)7=-3432  *yVy. 

EXERCISE 
609.  ^Vwc?  o^/y  £/ie  terms  asked  for : 

1.  The  third  term  of  (a  -b)9. 

2.  The  third  term  of  (-  a  +  z)10. 

3.  The  third  term  of  (a  +  Z>)100. 

4.  The  sixth  term  of  (x2  —  or1)8. 

5.  The  seventh  term  of  (2  ar*  -y-*)6. 

6.  The  middle  term  of  (a;  +  -\ * 

7.  The  middle  term  of  (3  x  -  $)». 

8.  The  sixth  term  of  (1  +  x)\ 


The  Binomial  Formula  447 

9.    The  two  middle  terms  of  f  x )  . 

\        x) 

10.  The  term  containing  a,*5  in  (a  —  x)7. 

11.  The  term  containing  a5  in  (a  —  x)9. 

12.  What  term  of  what  power  of  a  -f  b  contains  a?bn  ?  a?b11  ? 

13.  Write  the  first  four  terms,  and  the  last  term  of  (2  a2—  b%)8. 

14.  Find,  in  simplest  radical  form,  the  value  of  (V2  +  V3)4. 

Write  the  first  three  terms,  and  the  last  term  in  the  following  : 

15.  (2  a* -by.  19.    f-*-<2iA\ 

16'    (I-3)*  20.    (-*a-§by. 

17.  (1- fz)6.  21.    (3-2a>)». 

18.  (— |a;-f  fy)5.  •     22.    (-3  +  2z)8. 

23.  Find  the  7th  and  8th  terms  of  (a  +  6)10. 

24.  Find  the  4th  term  of  (a  +  6)11.     The  4th  from  the  last. 

25.  Write  the  next  to  the  last  term  of  (3  a*  —  b*)w. 

26.  What  is  the  exponent  of  x  in  the  first  term  of  (x  -f  ar1)12  ? 
in  the  second  term  ?  in  the  third  term  ?  Find  the  term  that 
does  not  contain  x. 

27.  Is  there  a  term  in  (x  -f  a?-1)11  that  does  not  contain  x  ? 

28.  Find  the  last  three  terms  of  (V2  —  b?f. 


XXVII-    VARIATION 

610.  In  numbers  that  are  related  to  each  other  through 
mathematical  equations,  some  of  the  numbers  may  be  changing 
in  value,  while  others  may  have  fixed  values. 

If  a  train  travels  at  a  uniform  rate  of  r  miles  per  hour,  we 
may  express  the  distance  it  has  traveled  after  the  lapse  of 
any  time  by  the  equation,  d  =  rt.  In  this  equation  d  and  t 
vary  in  value  from  moment  to  moment,  but  r  is  a  constant, 
for  by  the  conditions,  the  rate  is  uniform. 

611.  Variable  and  Constant.  A  number  that  is  changing  in 
value  is  a  variable;  a  number  whose  value  does  not  change 
is  a  constant. 

The  formulas  of  algebra,  geometry,  physics,  and  their  prac- 
tical applications,  involve,  in  general,  variables  and  constants. 

In  the  illustration  just  given,  d  =  rt,  d  and  t  are  variables  and  r  is  a 
constant. 

In  A  =  7ri?'2,  A  and  B  are  variables  and  ir  is  a  constant. 

612.  Direct  Variation.  If  two  variable  numbers  are  so  related 
to  each  other  that  through  all  their  changes  in  value  their  ratio 
remains  unchanged,  one  of  these  numbers  varies  directly  as  the 
other,  or  simply  varies  as  the  other. 

613.  Constant  of  Variation.  The  constant  value  of  the  ratio 
of  the  variable  numbers  in  direct  variation  is  the  constant  of 
variation. 

We  may  write  d  =  rt,  when  r  is  constant  and  d  and  t  are  variables,  in 

the  form  -  =r.      Then  by  definition,  we  have  "distance  varies  as  the 

t 
time." 

The  student  must  remember  that  d,  t,  and  r  are  abstract  numbers. 
They  represent  the  numerical  measures  of  concrete  magnitudes  ;  that  is, 
d  equals  the  number  of  miles  traveled  in  t  hours. 

448 


Variation  449 

614.  Notation.  The  symbol  for  variation  is  oc.  a  oc  b  is  read 
"  a  varies  as  b" 

It  is  customary  to  use  a  letter  with  different  subscripts  to 
represent  different  values  that  a  variable  number  has  at  dif- 
ferent periods  of  its  variation. 

Thus,  di,  d2,  ds  •■•  (read  d-sub  one,  etc.)  are  used  to  represent  the  dis- 
tances traveled  in  the  times  ti,  t*.,  h  •••,  respectively. 

615.  In  agreement  with  this  notation  and  the  definition  of 
variation,  we  have,  for  uniform  motion, 

*  =  ,,  d2=       ds=        t(, 

h  to  tZ 

where  r  is  the  constant  of  variation,  in  this  case  the  uniform 
rate  of  motion. 

It  is  evident  that  the  constant  of  variation  can  be  found  in 
any  particular  case,  if  we  know  a  set  of  corresponding  values 
of  the  two  variables. 

Thus,  if  d\  =  140  miles,  and  t\  —  4  hours,  r  =  ^^  =  35,  the  number  of 
miles  per  hour. 

616.  If  a  and  b  are  two  variable  numbers,  and  accb,  then 
ax :  a2  =  bx :  b2,  where  cti  and  blt  a*,  and  b2  are  sets  of  corre- 
sponding values  of  the  variables. 

Proof.     We  have  given  accb. 

.-.  ^-=k,  and  ^1—  k,  where  k  is  the  constant  of  variation. 

&1  &2 


al_a2 
bx      b2 

.  fit— 5s. 

<h      b2 


(Why?) 
(Why?) 


617.  On  account  of  the  possibility  of  expressing  the  rela- 
tion of  variation  in  the  form  of  a  proportion  as  just  proved,  it 
is  common  to  speak  of  one  of  two  variable  numbers  as  propor- 
tional to  the  other.     This  means  that  one  varies  as  the  other. 


450  Variation 

This  form  of  expressing  the  relation  of  variation  is  often  used 
in  geometry  and  in  physics. 

Thus,  in  geometry  we  say,  "  The  areas  of  triangles  having  equal  bases 
are  proportional  to  their  altitudes."  In  physics  for  uniform  motion,  we 
have,  "The  distance  is  proportional  to  the  time." 

EXERCISE 

618.  1.  If  x  oc  y,  and  x  =  10,  when  y  =  5,  what  is  the  constant 
of  variation  ? 

2.  Using  the  data  and  answer  of  example  1 ;  find  yx  if 
xl  =  50. 

3.  The  area  of  a  circle  varies  as  the  square  of  the  radius. 
Express  this  as  a  variation,  using  A  and  R  for  area  and  radius 
respectively. 

4.  Ax  =  314.16,  Rx  =  10.  Find  the  constant  of  variation 
in  example  3. 

5.  Find  R2,  if  A2  =  100. 

6.  The  area  of  a  triangle  varies  as  the  product  of  its  base 
and  altitude.     What  is  the  constant  of  variation  ?    ( T  =  ±  bh.) 

7.  From  example  6  show  that  7\  :  T2  =  &i  •  hi :  b2  •  h2,  where 
T,  b,  and  h  represent  respectively  the  area,  the  base,  and  the 
altitude  of  a  triangle. 

8.  For  a  freely  falling  body  we  have,  in  physics,  the 
formula  S  =  ^  gt2,  where  g  is  a  constant.  Show  that,  for  fall- 
ing bodies,  the  distance  is  proportional  to  the  square  of  the 
time. 

9.  From  example  8  derive  £]  :  $2  =  t^  :  t22. 

10.  Given  S1  =  64  feet  and  tx  =  2  seconds.     Find  g. 

11.  The  weight  of  a  sphere  of  given  material  varies  (di- 
rectly) as  the  cube  of  the  radius.  Two  spheres  of  the  same 
material  have  radii  of  2  inches  and  6  inches  respectively.  If 
the  weight  of  the  first  is  6  pounds,  what  is  the  weight  of 
the  second  ?  (Sheffield  Scientific  School.) 


Variation  451 

12.  Find  from  example  11  the  radius  of  a  sphere  that 
weighs  48  pounds. 

13.  The  surface  of  a  sphere  varies  as  the  square  of  the 
radius.  Express  this  in  the  form  of  a  variation.  Express  as 
a  proportion. 

14.  If  the  surface  of  a  sphere  is  1256  square  inches  when 
the  radius  is  10  inches,  what  is  the  constant  of  variation  ? 

15.  If  x  oc  y  and  y  oc  z,  show  that  x  oc  z. 

Let  k,  I,  and  m  be,  respectively,  the  constants  of  variation.  Show  that 
kl  =  m. 

619.  Inverse  Variation.  One  number  varies  inversely  as  an- 
other if  the  ratio  of  the  first  to  the  reciprocal  of  the  second  is 
constant. 

Thus,  x  varies  inversely  asy,  if  x  =  k  •  -,  or  xy  =  k. 

y 
The  inverse  variation  of  x  and  y  may  be  indicated  by  any 

1  7c 

one  of  the  three  expressions,  x oc-,  x  =  -,  or  xy  =  k.     Here  k 

is  the  constant  of  variation.  &  ^ 

Instead  of  "  varies  inversely "  we  sometimes  say  "  is  in- 
versely proportional  to." 

EXERCISE 

620.  1.  If  a;  varies  inversely  as  y,  and  xl  and  yx  are  respec- 
tively 9  and  2,  what  is  the  constant  of  variation  ? 

2.  Find  y2  in  example  1,  if  x2  =  30. 

3.  If  x  varies  inversely  as  y,  show 
that  Xi :  x2  =  y2  •  y*. 

4.  Show  that  any  two  altitudes  of  a 
triangle  are  inversely  proportional  to 
the  sides  upon  which  they  are  drawn, 
using  the  relation  T  =  \  aha  =  |  bhb. 

5.  If  x  varies  inversely  as  y,  show  that  for  a  multiplica- 
tion of  the  value  of  y  by  any  number  we  have  a  division  of 
the  value  of  x  by  the  same  number. 


452  Variation 

6.  A  train  has  a  run  of  240  miles.  Show  that  the  time 
required  is  inversely  proportional  to  the  rate  of  the  train. 

7.  A  person  has  a  given  sum  of  money  with  which  to  buy 
horses.  Does  the  number  of  horses  that  he  can  buy  vary 
directly  or  inversely  as  the  price  per  horse  ? 

8.  A  certain  piece  of  work  is  to  be  done.  Does  the  time 
required  to  do  the  work  vary  directly  or  inversely  as  the 
number  of  workmen  employed  ? 

9.  If  x  oc  -  and  y  cc  -,  show  that  x  oc  z. 

10.  The  weight  of  a  body  varies  inversely  as  the  square  of 
its  distance  from  the  center  of  the  earth.  If  a  body  weighs 
1  pound  on  the  surface  of  the  earth  (4000  miles  from  the 
center),  how  much  will  it  weigh  10,000  miles  from  the  center  ? 

621.  Joint  Variation.  If  one  number  varies  as  the  product 
of  two  others  it  varies  jointly  as  these  two  other  numbers. 

622.  If  x  varies  as  y  when  z  is  constant,  and  x  varies  as  z  when  y  is 
constant,  then  x  varies  jointly  as  y  and  z. 

Proof.  Let  xu  yu  zx  and  x2,  y2,  z2  be  any  two  sets  of  cor- 
responding values  of  the  three  variables. 

Consider  the  variation  of  y  and  z  as  taking  place  separately, 
and  let  x  change  in  value  from  xl  to  x'  due  to  the  change  in  y 
from  2/1  to  y2,  z  remaining  constant. 

.-.  *1  =  &.    (§616.)  (1) 

x'     y2 

Now  let  y  remain  constant  and  z  change  from  2!!  to  z2,  which 
will  change  x  from  the  intermediate  value  x'  to  x2. 

(2) 

r  =  rr-  (3) 

(4) 


.  x  . 

=  z-i. 

x2 

*2 

«a. 

=  .ViZi 

X2 

2/2z2 

JSl, 

=  _x2_ 

yi*i 

2/2*2 

Variation  453 

This  last  equation  shows  that  the  ratio  of  x  to  yz  is  the  same 
for  any  two  sets  of  corresponding  values  of  x,  y,  and  z. 

That  is,  —  =  fc, 

yz 

or  x  oc  yz. 

623.  If  x  varies  directly  as  y  and  inversely  as  z,  then  x  «- • 

Let  the  student  prove  this. 

It  is  evident  that  if  xcc  &,  we  have  an  increase  in  x  for  an 

2! 

increase  in  yy  but  a  decrease  in  x  for  an  increase  in  z. 

EXERCISE 

624.  1.  If  a;  varies  jointly  as  y  and  z,  and  a^  =  63,  yY  =  5, 
^  =  9 ;  find  the  constant  of  variation. 

2.  By  using  the  constant  of  variation  found  in  example  1, 
find  y2,  if  x2  =  72  and  z2  =  18. 

3.  Given  that  xcc  jointly  as  y  and  z,  and  a^  =  225,  yt  =  12, 
zx  =  15,  x.2  =  405,  ?/2  =  .6 :  find  by  proportion  the  value  of  z2. 
(See  equation  3  of  §  622.) 

4.  The  total  area  T  of  a  right  circular  cylinder  varies 
jointly  as  R  and  R-\-  H,  where  R  is  the  radius  of  the  base  and 
H  is  the  altitude.  When  R  =  7  inches  and  H=  13  inches, 
T  =  880  square  inches  ;  find  T,  when  R  =  5  inches  and  i7  =  10 
inches. 

5.  The  weight  of  right  circular  cylinders  of  the  same 
material  varies  jointly  as  the  height  and  the  square  of  the 
radius  of  the  base.  A  steel  cylinder  weighing  22  pounds  has 
a  base  with  radius  1  inch  and  its  altitude  is  7  inches.  Find  the 
weight  of  another  cylinder  whose  base  has  a  radius  of  2  inches 
and  whose  altitude  is  14  inches. 

6.  The  time  required  by  a  pendulum  to  make  one  vibration 
varies  directly  as  the  square  root  of  the  length.  If  a  pendulum 
100  centimeters  long  vibrates  once  in  a  second,  find  the  time 
of  one  vibration  of  a  pendulum  36  centimeters  long.     (Yale.) 


XXVIIL  LOGARITHMS 


625.  The  processes  of  multiplication,  division,  involution, 
and  evolution  can  be  greatly  abridged  by  the  use  of  the  laws 
of  exponents.  A.  system  of  computation  by  means  of  tables 
is  based  upon  these  laws. 


By  means  of  a  table  of  powers  of  2  we  can  perform  the 
operations  of  multiplication,  division,  involution,  and  evolution 
upon  powers  of  2. 

ORAL   EXERCISE 

627.   1.   32x128  =  ? 
Solution 


20  =  1. 
2*=  2. 

22  =  4. 

23  =  8. 
2*  =  16. 


26  =  64. 

27  =  128. 

28  =  256. 

29  =  512. 
210  =  1024. 
2ii  =  2048. 

212  -  4096. 

213  -  8i92. 
2i4  =  16384. 

215  =  32768. 

216  =  65536. 
2"  =  131072. 

218  =  262144. 

219  =  524288. 

220  =  1048576. 


From  the  table  32  =  25  and  128  =  27. 
.-.  32  x  128  =  25  x  27  =  25+7  =  212  =  4096. 

2.   163847=? 
Solution.    From  the  table  16384  =  21*. 

.-.  16384*  =  (2U)7  =  24 


16. 


3. 
4. 
5. 
9. 
10. 


256  x  8  =  ? 
642  =  ? 


V1024  =  ? 


6.  V4096  =  ? 

7.  (32768  X  8192)f  =  ? 

8.  J/6EEm  =  ? 


Divide  1048576  by  2048. 
Divide  524288  by  512. 


11.   Divide  8192  by  S/1024. 


628.  Logarithm.  The  logarithm  of  a  num- 
ber is  the  exponent  of  the  power  to  which  a 
fixed  number  called  the  base  must  be  raised  to  produce  the 
number. 

454 


Logarithms  455 

Thus,  in  213  =  8192,  13  is  the  logarithm  of  8192  to  the  base  2.     This, 
in  the  notation  of  logarithms,  is  written 

log2  8192  =  13, 
and  is  read,  the  logarithm  of  8192  to  the  base  2  is  13. 

Any  expression  of  the  form  ab  =  c  can  be  changed  to  loga- 
rithmic notation. 

Thus,  ah  =  c  and  log0  c  =  b  according  to  the  definition  of  logarithm, 
represent  the  same  relation  between  a,  6,  and  c. 

EXERCISE 

629.    Change  the  following  into  logarithmic  notation : 

1.  23  =  8.  4.   64*  =  16.  7.   9-^  =  1 

2.  7*  =  14.  5.    64- =  16.  8.   8*  =  32. 

3.  mk  =  y.  6.   3*^  =  27.  9.   ^  =  32. 

Bead  the  following,  and  change  each  from  the  logarithmic  nota- 
tion to  the  exponential  form : 

10.  log6a=c.  15.  log981  =  2. 

11.  loga6  =  c.  16.  log!  81  =-2. 

12.  log64l6  =  .|.  17.  log9a;  =  f 

13.  logaa3  =  3.  18.  log3729  =  7. 

14.  logl0l  =  0.  19.  logx32  =  5. 

Find  the  value  of  x  in  each  of  the  following : 
20.    Solve  for  x,  log10 100  =  x.       27.    log16  8  =  x. 

28.  loga  a  =  x. 

29.  loga  1  =  x. 

30.  logo  8= x +1. 

31.  log327=flj. 

32.  logo2a8  =  a2. 

33.  log2  x  =  5. 
Solution.     25  =  x. 

.-.  x  =  32. 

34.  loga2a  =  -  2. 

35.  log2  x  =  8. 


Sol 

DTION.       10*  =  100. 

But 

102  =  100. 

.-.  10*  =  102. 

.-.  x  =  2. 

21. 

log232=Z. 

22. 

log32  2  =  x. 

23. 

loga  a4  =  x. 

24. 

log2 1  =  X. 

25. 

log.5 .125  =  x. 

26. 

log8 1 6  =  a. 

456  Logarithms 

630.  Laws  of  Logarithms.  The  laws  of  logarithms  for  multi- 
plication, division,  involution,  and  evolution  are  exactly  the 
same  as  the  corresponding  laws  of  exponents,  as  the  student- 
might  anticipate,  since  logarithms  are  exponents. 

1.  Law  of  Multiplication.  The  logarithm  of  a  product  equals  the 
sum  of  the  logarithms  of  its  factors. 

In  symbols,         logkab  =  logka  -f  \ogkb. 

Proof.         Let  log,  a  =  x  and  log,  b  =  y. 

...  kx  —  a  and  kv  =  b.     (Definition  of  logarithm.) 
...  ]{*+»  =  ab.     (Why  ?) 
.*.  logkab  =  x  -f-  y,     (Definition  of  logarithm.) 
or  log,  ab  =  log*  a  +  log,  b.     (Why  ?) 

2.  Law  of  Division.  The  logarithm  of  a  quotient  equals  the  logarithm 
of  the  dividend  minus  the  logarithm  of  the  divisor. 

In  symbols,  logfe^  =  logka  -  \ogkb. 

Proof.         Let  log*  a  =  x  and  log,  b  =  y. 

.-.  kx  =  a  and  ky  =  b.     (Why  ?) 

...  kx~*  =  --     (Why?) 

.vlofof-*^*     (Why?) 
b 

or  log,  I  =  log,  a  -  log,  b.     (Why  ?) 
b 

3.  Law  of  Powers.  The  logarithm  of  the  power  of  a  number  equals 
the  exponent  of  the  power  multiplied  by  the  logarithm  of  the  number. 

In  symbols,         log^o**  =  n  •  logfca. 

Proof.         Let  log,  a  =  x. 

.-.  Af  =  a.      (Why?) 
...  kn*  =  a\     (Why  ?) 
.-.  log,  an  =  nx,    (Why  ?) 
or  log,an  =  n  log,  a.     (Why  ?) 


•  Logarithms  457 

4.  Law  of  Roots.    The  logarithm  of  the  root  of  a  number  equals  the 
quotient  of  the  logarithm  of  the  number  divided  by  the  index  of  the  root. 

In  symbols,       log^ Vfl  =  -  logfca. 

n  < 

Proof.         Let  log*,  a  =  x. 

.-.  7<r  =  a.     (Why?) 

.-.  kn=Va.     (Why?) 

/.  log*  -y/a  =  -•#  =  -•  log*  a. 
n  n 

631.  According  to  these  laws  we  may  make  such  transfor- 
mations as  the  following : 

1.  log  —  =  log  a  +  log  b-  log  c.     (Why?) 

c 

2.  log(a&)2  =  2(loga&)=21oga  +  21og&.     (Why?) 

3.  |log*-|lQgy+|log*=log^^.     (Why?) 

4.  31oga-41og6  =  log^.     (Why?) 

EXERCISE 

632.  Using  the  laws  of  logarithms  express  examples  1  to  9  in 
terms  of  log  a,  log  b,  log  c,  and  log  x  as  in  examples  1  and  2,  §  631. 

1.  log  3  a*x.  5.   log  a  Vb. 

2.  log  («\\  6.   log  -Vab. 

3.  !<*(<* +  b*).  7-   lo^^_ 

Up  8.   log  aWbxc". 

4.  loga\/— •  . 

^c  9.   logVaa2. 


458  Logarithms 

Express  examples  10  to  14  as  the  logarithm  of  a  single  term  as 
in  examples  3  and  4,  §  631. 

10.  log  a  +  log  b  —  log  c.  12.    3  log  a  +  4  log  6. 

11.  log  a  +  log  6  -  2  log  c.       13.   log  a2  +  2  log  b  -  2  log  aft. 

14.  log  -  +  log  xy  —  log  #  +  log  y. 

15.  log  (a -2/)  + log  (a +  ?/). 

633.  Common  Logarithms.  Any  other  base  than  2  might 
have  been  used  and  a  table  similar  to  that  of  §  626  formed. 
In  practice,  logarithmic  computations  are  made  with  the  com- 
mon or  Briggs  system  of  logarithms.  In  this  system  the  base 
is  10. 

Common  logarithms  are  exponents,  positive  or  negative,  of 
powers  of  10. 

From  the  definition  of  common  logarithms  since 


104    =  10,000,  . 

\  logio  10,000  =  4. 

103  =iooo,    . 

\  logio  1000     =  3. 

102    =100,       . 

•.  log10  100       =  2. 

101    =10, 

•.  log10  10         =  1. 

100    =1, 

•.  logio  1           =0. 

10-i  =.1, 

•.  logio  .1         =-1. 

io-2  =  .01, 

\  log10.01        =  -2. 

10-3  =  .001,      . 

•.logio  .001      =-3. 

Clearly  the  logarithms  of  numbers  between  1000  and  10,000 
lie  between  3  and  4 ;  similarly  the  logarithms  of  numbers  be- 
tween 100  and  1000  lie  between  2  and  3,  etc.  Therefore,  the 
logarithms  of  most  numbers  will  have  an  integral  part  and  a 
decimal  part. 

634.  Characteristic,  Mantissa.  The  integral  part  of  a  loga- 
rithm is  the  characteristic  and  the  decimal  part  is  the  mantissa. 
The  mantissa  is  always  positive. 

Thus,  logio  20  =  1.3010.  The  characteristic  of  logi020  is  1  and  the 
mantissa  is  .3010. 


Logarithms  459 

635.  Mantissa  Law.  The  mantissa  depends  only  upon  the  sequence 
of  the  figures  and  is  independent  of  the  position  of  the  decimal  point. 

Illustration.     We  may  find  in  the  tables  that 
logio625  =2.7959. 

log™  62.5  =  logio  625  -  log™  10  (By  law  2) 
=  2.7959  -  1 
=  1.7959. 

Note  that  the  mantissa  is  the  same  for  log  625  and  log  62.5. 

Proof.  Moving  the  decimal  point  to  the  right  or  the  left 
multiplies  or  divides  the  number  by  10  or  100  or  1000,  etc. 
Therefore  the  logarithm  of  the  number  will  be  increased  or 
diminished  by  log  10,  or  log  100  or  log  1000,  etc.  But  the 
logarithms  of  10,  100,  1000,  etc.  are  integral  numbers,  and  in- 
creasing or  diminishing  the  logarithms  by  integers  will  not 
change  the  decimal  part,  the  mantissa,  of  the  logarithm. 

636.  Characteristic  Law.  The  characteristic  of  a  number  greater 
than  unity  is  one  less  than  the  number  of  figures  to  the  left  of  the  deci- 
mal point. 

Consider  225.16.  As  it  lies  between  100  and  1000,  its  loga- 
rithm is  between  2  and  3 ;  that  is,  its  logarithm  is  2  +  a  frac- 
tion.    Similarly  for  any  number  more  than  unity. 

Find  between  what  two  powers  of  10  each  of  the  following 
lies :  (a)  21 ;  (b)  3.1 ;  (c)  5437.1.  What  is  the  characteristic 
of  the  logarithm  of  each? 

The  characteristic  of  a  number  less  than  unity  is  negative,  and  is 
numerically  equal  to  one  more  than  the  number  of  zeros  preceding  the 
first  significant  figure  of  the  number. 

The  following  will  illustrate  the  laws  that  govern  both 
characteristic  and  mantissa : 

logi07235  =3.8594.  (From  tables.) 

logio  723  5  =  logio  7235  -  log10  10  =  2.8594.  (  Why  ?) 

logio  72.35  =  log10  723.5  -  logio  10  =  1.8594.  (Why  ?) 

logio  7.235  =  log10  72.35  -  logio  10  =  0.8594.  (Why  ?) 

logio  -7235  =  logio  7.235  -  log10 10  =    .8594  -  1.  (Why  ?) 


460  Logarithms 

The  last  number  is  a  negative  number.  We  ordinarily 
WIlte  logw  .7235  =  1.8594. 

This  is  to  be  understood  to  mean  -  1  +  .8594.  The  mantissa  is  posi- 
tive. In  practice,  to  avoid  a  negative  characteristic  10  is  added  and  sub- 
tracted,thus,  1.8594  =  9.8594-10. 

Also  logio  .07235  =  log10  .7235  -  log10  10 
=  2.8594  =  8.8594-10, 
and  logio  .007235  =  3.8594  =  7.8594  -  10. 


EXERCISE 

637.  What  is  the  characteristic  of  the  logarithm  of  each  of  the 
following  numbers  ? 

1.  25.  3.    .0004.  5.    101.  7.    .9. 

2.  .5.  4.    1.732.  6.    99.  8.    .99. 

9.   If  log  247  =  2.3927,  what  are  the  logarithms  of  24.7? 
.0247  ?  2.47  ?  24,700  ?  .247  ?  .000247  ? 

10.  What  are  the  logarithms  of  3,  27,  81,  243,  and  -J-  in  a 
system  of  which  the  base  is  3  ? 

11.  What  are  the  logarithms  of  5,  25,  125,  625,  3125,  i  in  a 
system  of  which  the  base  is  5  ? 

12.  What  are  the  logarithms  of  36,  216,  1296,  in  a  system 
of  which  the  base  is  —  6  ?  Why  is  a  negative  number  not 
convenient  as  the  base  of  a  system  of  logarithms  ? 

Given  log  2  =  .3010,  log  3  =  .4771,  log  5  =  .6990,  find  : 

13.  log  6.  16.   log  15.  19.   log  7.5. 

14.  log  9.  17.    log34.  20.   logf. 

15.  log  12.  18.    log62.  21.    log  375. 

22.  How  many  figures  are  there  in  2530  ?  in  3025  ? 

23.  How  many  zeros  are  there  between  the  decimal  point 
and  the  first  significant  figure  of  (\)m  ?  (J)50  ? 

24.  How  can  you  find  log10  5  from  log10  2  =  .3010  ? 


Use  of  Tables  461 


USE  OF  TABLES 


638.  In  the  tables  on  pp.  462  and  463  the  mantissas  are 
given  correct  to  but  four  decimal  places.  By  using  these 
tables,  results  can  generally  be  relied  upon  as  correct  to  3 
figures  and  usually  to  4.  If  a  greater  degree  of  accuracy  is 
required,  five-place  or  even  seven-place  tables  must  be  used. 

639.  To  find  the  logarithm  of  a  given  number : 

Write  the  characteristic  before  looking  in  the  tables  for  the 
mantissa.     (§  636.) 

Find  the  mantissa  in  the  tables. 

(1)  When  the  number  consists  of  not  more  than  three  figures  : 

In  the  column  N",  at  the  left-hand  side  of  the  page,  find  the 
first  two  figures  of  the  number.  In  the  row  K,  at  the  top  or 
bottom  of  the  page,  as  convenient,  find  the  third  figure.  The 
mantissa  of  the  number  will  be  found  at  the  intersection  of 
the  row  containing  the  first  two  figures  and  the  column  con- 
taining the  third  figure. 

1.  Find  log  384. 

The  characteristic  is  2  (Why  ?).  In  the  column  N  find  38 
and  in  row  N  find  4.  The  mantissa  5843  will  be  found  at  the 
intersection  of  the  row  38  and  column  4. 

.-.  log  384  =  2.5843. 

2.  What  is  log  3.84  ?  log  38.4  ?  log  0.0384  ? 

(2)  When  the  number  consists  of  more  than  three  figures  : 

Find  as  above  the  mantissa  of  the  logarithm  of  the  number 
consisting  of  the  first  three  figures.  To  correct  for  the  remain- 
ing figures  interpolate  by  assuming  that,  for  differences  small  as 
compared  with  the  numbers,  the  differences  between  numbers  are 
proportional  to  the  differences  between  their  logarithms.  This 
statement  is  only  approximately  true,  but  its  use  leads  to 
results  accurate  enough  for  ordinary  computations. 


462 


Logarithms 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 
15 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 
20 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

3010 

3032 

3054  • 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

.3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 
25 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4106 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 
30 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 
35 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 
40 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

62(53 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

63(35 

6375 

6385 

6395 

6405 

6415 

6425 

44 
45 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

67130 

(5739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 
50 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Use  of  Tables 


463 


N 
55 

0 

1 

2 

3 

4 

5 

G 

7 

8 

9 

7404 

7412 

7419 

7427 

7435 

7143 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 
60 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 
65 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8.'344 

8351 

8357 

8363 

8370 

8376 

8382 

69 
70 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 
75 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 
80 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 
85 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9300 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

94(50 

9465 

9469 

9474 

9479 

9484 

9489 

89 
90 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

9542 

9547 

9552 

9557 

9562 

9506 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 
95 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98- 

9912 

9917 

9921 

9926 

9930 

99?>4 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

464  Logarithms 

Find  log  3847. 

Mantissa  of  log  3850  =    .5855. 
Mantissa  of  log  3840  =    .5843. 
10      0.0012. 
Mantissa  of  log  3847  =  .5843.+  ^  of  0.0012  =  .5851. 

The  difference  between  3840  and  3850  is  10  ;  the  difference  between  the 
mantissas  of  their  logarithms  (.5855  —  .5843)  is  0.0012.  Assuming  that 
each  increase  of  1  unit  between  3840  and  3850  produces  an  increase  of  1 
tenth  of  the  difference  in  the  mantissas,  the  addition  for  3847  will  be  7 
tenths  of  0.0012  or  0.00084.  .5843  +  0.00084  =  .5851.  Therefore,  the 
mantissa  of  log  3847  =  .5851. 

EXERCISE 

640.  Find  the  logarithms  of: 

1.    1845.  2.   6.897.  3.   0.04253. 

641.  To  find  the  number  corresponding  to  a  given  logarithm : 

The  number  corresponding  to  a  logarithm  is  its  antilogarithm. 
The  characteristic  determines   the  position  of  the  decimal 
point. 

(1)  If  the  mantissa  is  found  in  the  tables,  the  number  is  found 
at  once. 

Find  antilog  3.5877. 

The  mantissa  is  found  at  the  intersection  of  row  38  and 
column  7. 

\  antilog  3.5877  =  3870. 

(2)  If  the  exact  mantissa  is  not  found  in  the  tables,  the  first 
three  figures  of  the  corresponding  number  can  be  found  and  to 
them  can  be  annexed  figures  found  by  interpolation. 

Find  antilog  3.5882. 

log  3880  =  3.5888  log  required  number  =  3.5882 

log  3870  =  3.5877  log  3870 =  3.5877 

10      0.0011  log  req.no.  -  log  3870  =  0.0005 

3870  +  (A  of  10)  =  3874.54+. 


Use  of  Tables  465 

The  two  mantissas  in  the  table  nearest  to  the  given  mantissa  are  .5888 
and  .5877,  differing  by  0.0011.  The  corresponding  numbers,  since  the 
characteristic  is  3,  are  3880  and  3870,  differing  by  10.  The  difference 
between  the  smaller  mantissa  5877  and  the  required  mantissa  5882  is 
0.0005.  Since  an  increase  of  11  ten  thousandths  in  mantissas  corresponds 
to  an  increase  of  10  in  the  numbers,  an  increase  of  5  ten  thousandths  in 
mantissas  may  be  assumed  to  correspond  to  an  increase  of  T5T  of  10  in  the 
numbers.  Therefore  the  number  is  3870  +  (^  of  10)=  3874.54+.  The 
last  two  figures  are  uncertain. 

EXERCISE 

642.  Find  the  antilogarithms  of: 

1.  2.9445.  3.    1.6527.  5.    1.9994. 

2.  2.4065.  4.   3.7779.  6.   0.7320. 

643.  The  cologarithm  of  a  number  is  the  logarithm  of  its 
reciprocal.     The  cologarithm  of  100  equals  the  logarithm  of 

rfo>>  that  is>  ~  2- 

Since  log  1  —  0,  .-.  log-  =  log  1  —  log  n  =  0  —  log  n, 
n 

therefore  colog  n  =  —  log  n. 

As  the  cologarithm  of  a  number  equals  the  logarithm  with  its 
sign  changed,  adding  the  cologarithm  will  give  the  same  result 
as  subtracting  the  logarithm.  The  former  is  sometimes  more 
convenient. 

To  avoid  negative  results  it  is  often  more  convenient  to  add 
and  subtract  10. 

1.  Find  colog  47.3. 

In    subtracting    1.6749    or    any 

log  1  =  10.0000  —  10         other  logarithm  from  10,  the  result 

log  47.3  =    1.6749  may  be  obtained  mentally  by  sub- 

colog  47.3  =     8.3251  —  10         tracting  the  right-hand  figure  from 

10  and  all  the  others  from  9. 

2.  Find  the  value  of  4 


5371  x  29 


466  Logarithms 

4.PC9  v  93 

l0g  5371x29  =  l0g  452  +  l0g  23  ~  l0g  5371  ~  l0g  29 

=  log  452  +  log  23  -J-  colog  5371  +  colog  29. 
log  452  =  2.6551. 
log  23  =  1.3617. 
colog  5371  =  6.2699  -  10. 
colog  29  =  8.5376  -  10. 
Adding      8.8243  -  10. 
antilog  8.8243  -  10  =  0.066728+ 

Theref0le       5^1  =  °-066728+: 

3.    Find  50*. 

log  50?  =  flog  50. 
log  50  =  1.6990. 
|  log  50  =  |  of  1.6990  =  1.2742. 
antilog  1.2742  =  18.80. 

.-.  50*  =  18.80. 

644.  Compound  Interest.  Problems  in  compound  interest 
that  involve  long  computations  can  readily  be  solved  by  means 
of  logarithms. 

To  find  the  amount  (4)  at  the  end  of  n  years  of  a  given  sum 
of  money  (P)  invested  at  compound  interest  at  a  given  rate  (r) : 

The  amount  of  P  dollars  at  compound  interest,  at  the  end 
of  the  first  year  is,         A  —  P  +  rP  =  P(l  +  r). 

At  the  end  of  the  second  year, 

A  =  P(l  +  r)+  rP(l  +  r)  =  P(l  +  rf. 
At  the  end  of  the  third  year, 

A  =  i?(l  +  r)2  +  rP(l  +  r)2  =  P(l  +  r)3. 
At  the  end  of  the  nth  year, 

A  =  P(l  +  r)»-i  +  rP(l  +  r)»~i  =  P(l  +  r)». 


Use  of  Tables  467 

What  will  be  the  amount  of  $  1500  for  12  years  at  4  %,  the 
interest  being  compounded  annually  ? 

Here  A  =  1500(1  +  .04)" 

log  A  =  log  1500  4- 12  log  1.04 
=  3.1761  +  12  x  .0170 
=  3.3801. 
.-.  A  =  $  2405.55. 

EXERCISE 

645.  1.  Find  from  the  tables  the  logarithm  of  each  of  the 
following  numbers:  (a)  74;  (6)  129;  (c)  2004;  (d)  16.21; 
(e)9.547;      (/).018;      (g)  .21  j      (ft)  ft;      (i)  ft;      (j)  7|; 

»&■ 

2.  Find  the  logarithm  of  each  of  the  following  numbers : 
(a)  75;  (b)  21214;  (c)  3.1714;  (d)  31.2;  (e)  918.4;  (/)  .00084; 
to)  42.53;  (ft)  .18713;  (i)  .00427. 

3.  Find  from  the  tables  the  numbers'  corresponding  to  the 
following  logarithms:  (a)  .7412;  (b)  2.9983;  (c)  .9060; 
(d)  .7033 ;    (e)  4.9883 ;    (/)  1.0881 ;    (gr)  3.6538 ;    (ft)  3.5051. 

4.  Perform  the  following  operations  by  means  of  loga- 
rithms :  (a)  256  x  311 ;  (6)  451  X  215  ;  (c)  7643  -r-  213 ; 
(d)  972  +  41 ;     (e)  158  x  ^39  ;     (f)  74  x  411 ;     (g)  615  x_53 ; 

(ft)  613  -h  174 ;      (I)  193  x  810 ;      (J)  17  V29  ;      (ft)   41  '1^613  i 

(0  36«  x  (A)*- 

5.  At  birth  a  child  has  $  500  placed  in  the  bank  for  him, 
to  accumulate  at  4  %  compound  interest  till  he  is  21.  What 
amount  will  he  receive  when  he  is  21  ? 

6.  The  first  Folio  of  Shakespeare,  regarded  as  the  most 
valuable  book  printed  in  the  English  language,  was  published 
in  1623.  The  original  cost  was  £1  or  approximately  $5. 
The  last  copy  offered  for  sale  in  1912  brought  $  9000.     One 


468  Logarithms 

would  naturally  think  that  the  purchaser  of  this  first  Folio  in 
1623  made  a  fine  investment.  What  would  an  original  invest- 
ment of  $  5  amount  to  in  1912  at  6  %  compound  interest  ? 

7.  If  at  the  beginning  of  the  year  1,  one  cent  had  been 
invested  at  4  %  compound  interest,  what  would  the  amount  be 
in  1915  ?  What  would  be  the  radius  of  a  sphere  of  gold  that 
would  represent  the  value  of  the  investment  in  1916,  if  a  cubic 
foot  of  gold  is  worth  $  362,900  ? 

8.  If  log  2= .3010  find  the  value  of  x  in  the  equation  2X= 10. 

9.  Compute  the  value  of  3  2  by  means  of  logarithms. 

(Harvard.) 

10.  About  300  years  ago  the  Indians  sold  Manhattan  Island 
to  Peter  Minuit  for  $  24.  Suppose  this  money  had  been  put 
out  at  compound  interest  at  6%,  how  much  would  it  have 
amounted  to  at  the  present  time  ? 

11.  According  to  the  will  of  Benjamin  Franklin,  the  cities 
of  Boston  and  Philadelphia  each  received  £  1000  in  July  1791 
to  be  invested  at  5  %  compound  interest  for  100  years.  In 
July  1891  the  total  amount  of  the  fund  in  Boston  was 
$  391,168.68  and  in  Philadelphia  $  100,000.  How  much  should 
have  been  realized  by  the  terms  of  the  will  ?    (£  1000=  $  5000.) 

12.  A  chain  of  letters  is  started  for  the  purpose  of  aiding 
an  old  railroad  man  who  is  ill.  Number  1  sends  a  letter  to 
each  of  5  friends,  each  of  them  in  turn  sends  a  letter  to  5 
friends,  and  so  on.  If  the  chain  ends  with  letter  number  50 
and  each  person  who  receives  a  letter  sends  10  cents,  how 
much  does  the  man  receive  ? 


XXIX.  GENERAL  REVIEW 

646.  1.  If  a  =  3,  b  =  2,  c  =  1,  find  the  value  of  each  of  the 
following  expressions : 

(1)  2  a2  -  b2 ;  2(a2  _  52) ;  (2 a*  _  &*)2.  2(a2  -  62)2. 

(2)  a&c-(a  +  6  +  c).  (3)  (a2  +  b*)(a  +  6)(a  -  6). 
(4)  [a*+(b-c)a-bc](b-c).       (5)  V(a  +  6  +  c)  a&c. 

2.  Verify  the  following  identities  : 

(2)^  =  ^  +  9)2-^-?)2. 

(3)  1  +  2  +  3  +  4+.. >  +  n  =  n(?l2+  ^  » 

(4)  1  +  3  +  5+  ...  +(2n-l)=n2. 

3.  Solve  the  following  problems  by  translating  the  verbal 
language  of  the  problem  into  an  equation  with  one  unknown : 

(1)  In  five  years  a  boy  will  be  double  the  age  he  was  five 
years  ago.     How  old  is  he  ? 

(2)  I  have  as  many  brothers  as  sisters  said  a  boy.  And  I, 
said  one  of  his  sisters,  have  twice  as  many  brothers  as  sisters. 
How  many  brothers  and  sisters  were  there  ? 

(3)  Can  there  be  three  consecutive  integers  such  that  their 
sum  is  three  times  the  smallest  ? 

(4)  The  sum  of  three  consecutive  numbers  is  three  times  the 
middle  number.  What  are  the  three  numbers  ?  Does  this 
problem  lead  to  an  equation  or  to  an  identity  ? 

4.  Express  in  algebraic  language  the  following  theorems : 
(1)  Thcproduct  of  two  numbers  is  equal  to  the  difference 

of  the  squares  of  their  half  sum  and  their  half  difference. 

469 


470  General  Review 

(2)  Every  integer  that  is  a  perfect  square  diminished  by 
unity  is  equal  to  the  product  of  the  number  that  is  one  less 
than  its  square  root  by  the  number  that  is  one  more  than  its 
square  root. 

(3)  The  difference  between  the  squares  of  two  consecutive 
integers  is  an  odd  number,  obtained  by  increasing  by  unity 
twice  the  smaller  of  the  two  numbers. 

5.  The  lengths  of  the  sides  of  a  triangle  are  a  =  5  inches, 
6=4  inches,  c  =  3   inches.     Indicate  the  semi-perimeter  by 

s  =  ,   and   find     the    area,    A,   of    the    triangle,    if 


A  =  Vs(s  —  a)(s  —  b)(s  —  c). 

6.  If  2  s  represents  the  perimeter  of  .a  triangle  and  a,  6,  c 
its  sides,  verify  the  following : 

(1)  -  a  +  6  +  c  =  2(s  -  a).  (4)  a  =  2  s  -(b  +  c). 

(2)  a-b  +  c  =  2(s-b).  (5)  b  =  2s-(a  +  c). 

(3)  a  +  b  -  c  =  2(8  -  c).  (6)  c  =  2s  -(a  +  b). 

7.  Having  given  the  trinomials  r  +  8  +  t,  r  +  s  —  t,  r  —  s  +  t, 
—  r  -f  s  +  t,  from  the  sum  of  the  first  three  subtract  the  sum 
of  the  last  three  increased  by  the  sum  of  the  second  and  third. 

8.  If  A=(p  +  q)  +  (r  +  s),  B  =  (p  +  q)-(r  +  8),  C=(p-q) 
+  (r  —  s),  D=(p-q)-(r-s),  find  the  value  of  A  +  B+C 
+  D  and  of  A  x  D  by  type  multiplication. 

9.  Prove  that  [m  —  (p  +  q)+  r]  —  \m  —  [(p  +  q)—  r] \  -f 
\m  —  (p  +  r)+q\  =  m  —  p  +  q  —  r. 

10.  Apply  the  general  formulas  to  the  following : 

(1)  (tf-<?y.  (5)  (a  +  b-c)(a-b  +  c). 

(2)  (4  a2 -5  62)2.  (6)  (a2  +  a  +  l)(a*-  a  +  1). 

(3)  (2o-«)(2o+2i).  (7)  (w  +  2)(n-f-3)(w2+5n+6). 

(4)  (1  +  x)(x  -  l)(aj«  -  1).  (8)  1032. 

11.  Show  that  (1  +  x  +  a;2  +  &  +  ^  +  ^(l  -  »)=  1  -  x*. 
Show  also  that  (l-a^  +  ^-^  +  aJ4-  &)(!  +  »)=  1  -  a:6- 


General  Review  471 

12.  Divide  3  x*p  +  14  x?p  -f  9  xp  +  2  by  3  x2p  —  xp  4-  2. 

13.  Divide  a3™  —  x*n  by  cc2"1  +  xm+n  +  x2n. 

14.  What  value  must  the  coefficient  k  have  in  order  that 
x*  —  5  a3  4  9  a?2  4  kx  +  2  may  be  exactly  divisible  by  x2  —  3  x 
+  2? 

15.  Apply  the  general  formulas  to  the  following : 
(1)  (16  -  »*)-i-  (a*  +  4).  (2)  (1  -  a3)  -  (a  -  1). 

(3)  (w9  +  64  a6615)  -r-  (4  a265  +  ?i3). 

(4)  (mV4-l)-s-(mW  +  l). 

(5)  [^-(a-6)3]-(a;-a  +  &). 

(6)  [(j,  _  g)2  _  (r  _  S)2]  +  {p_q_r+  8). 

16.  Divide  a2(6  +  c)  —  62(c  +  a)-fc2(a  +  5)  +  abc  by  a  —  6 
+  c. 

17.  Square  and  cube  each  of  the  following  : 

2a  +  1;    a  — 7;    3#  —  o#  ;    asc2  +  6?/2;     a>4--Jj9;    }m-|w. 

18.  Square  each  of  the  following  : 

a3  —  a2 -|-a  —  1;    x  —  y  +  z  +  1;    ax2  -\-  bx  4  c. 

19.  Verify  the  following  identities  : 

(1)  (a2  +  62)2  =  (a2  -  62)2  +  (2  ab)2. 

(2)  (a2  4-  b2  +  c2)2  =  (a2  +  b2  -  c2)2  +  (2  ac)2  +  (2  6c)2. 

(3)  [(ti  4-  2)2  -  (n  +  l)2]  -  [(n  +  l)2  -  n»]  =  2. 

(4)  Show  that  the  identity  (n  4-  l)2  —  »2  =  2?*  4- 1  expresses 
that  the  difference  between  the  squares  of  two  consecutive 
integers  is  always  an  odd  number. 

20.  Evaluate  each  of  the  following  expressions  : 

(1)  2  x  5  4- 12 -f- 4- 7  +6-tAl. 

6 

(2)  6  x  7  -  32  x  5  4^8  X  5  -  7. 

21.  Extract  the  square  roots  of  the  following  polynomials  : 

(1)  3a2-2a  +  l-2a34-«4. 

(2)  16a8  f  9ys-S0x2f  +  4,9xY  —  i0x6y2. 


472  General  Review 

22.  Factor  into  prime  factors  : 

(I)  7  a4  +  7  a2b2  -  14  a3b.  (2)  m2(a  -  b)  +  n2(6  _  a). 

(3)  7pga;2  -  42_pga;  +  63  pg  -  7pra2  +  42prx  -  63pr. 

(4)  a;y  -  x  +  y  —  1.  (5)  xa'  +  a#"  +  x2  +  «V. 

(6)  a,*3  4-  ?/3  —  a;2?/  —  a,^2. 

(7)  a2b  +  62c  +  c2a  +  a2c  +  62a  +  c2b  4-  3  a&c. 

(8)  7  a;2 -28  a4. 

(9)  —  2uv—u2  —  v2—2uw  —  2vw  —  w2. 

(10)  x*(x2  -y2)-f  (x2  -  y2)  -xy(x-  y)2  (x  +  y). 

(II)  (a2  +  b2)2  -  (a2  -  62)2. 

(12)  x(x  -  1)  -  (x  -  l)2  +  &  -  1. 

(13)  a3  —  a?  4-  a2a;  —  aa;2  —  a  4-  a\ 

(14)  ^  _  aty.  (15)  ^  +  10  a^  +  21. 

(16)  5  a2a;4  -  5  a4a;2  -  5  a262a;2  4-  5  aAb2. 

(17)  a4  —  4  nx3  +  6  n2a;2  -4A  +  n4. 

(18)  3a3  -  16a;2  -  3a;  4- 16.  (22)  x2+  2ax  -  b2  +  a2. 

(19)  a;2  -  7  a;  +  10.  (23)  x*  -  3  a2a;2  +  a4. 

(20)  3  a;2  +  12  x  +  9.  (24)  a*  -  (a3  +  &3).^  +  a363. 

(21)  9  a;2  -  12  x  -  5.  (25)  ax2  +  (a  +  b)xy  +  by\ 
(26)  4(a6  4-  cd)2  -  (a2  +  62  -  c2  -  d2)2. 

23.  Solve  the  following  equations  by  factoring : 

(1)  x2-(a  +  b)x  +  ab  =  0.  (6)  a?  -  2a;2  -  4a;4-8  =  0. 

(2)  p2  +  3i)  +  2  =  0.  (7)  3a?  +  7a;2  =  3a;  4- 7. 

(3)  t*  -  t2  =  9 1  -  9.  (8)  v3  +  v2  -  v  -  1  =  0. 

(4)  z2  +  2  -  30  =  0.  .     (9)  s4  4-  s2  -  12  =  0. 

(5)  x2  4-  £a>  -  £  =  0.  (10)  &3  4-  k2  =  0. 

(11)  Find  a  number  such  that  if  3  and   5  are  subtracted 
from  it  in  turn,  the  product  of  the  two  remainders  is  120. 

(12)  Find  two  numbers  such  that  their  difference  is  2  and 
the  sum  of  their  squares  is  130. 

24.  Find  the  H.  C.  F.  of  a;2  -  3  a;  4-  2,  a;2  -  2  a;  4- 1,  x2  +  x-2. 

25.  Find  the  H.  C. F.  of  x2  +  2a;4- 1,  x4  -  10a;2  4-  9, 

x*  4-  2a?2  -5x  -6. 


General  Review  473 

26.  Find  the  L.  C.  M.  of  x2+(a  +  b)x  +  ab  and 

x2  +  (a  —  b)x  —  ab. 

27.  Find  tfie  L.  C.  M.  of  p3  +  q*,  p2  -  q2,  p2  +  2pq  +  q\ 

28.  Simplify  the  following  fractions : 

(x  +  a)2_(p  +  cy  30  a;2-  18a;-  12 

1  ;   (x  +  b)2  -(a  +  c)2'  U  16a;2  +  4a; -20  ' 

a;2_6a;H-5  ^  2a3  -  7a;2  +  7x  -  2 

{  }   x2  -11a;  +  10'  ^  '  3a?  -  10a;2  +  9a; -2* 

29.  Reduce  each  of  the  following  groups  of   fractions  to 
groups  having  a  common  denominator : 

(I)-,-,--  (2)      m  p 


a    b7  c  x  '   a +  6'  a-b'    a?-br 

(S)        P  +  1  P-1  l>-7 

V  ;  p2-8p  +  7'  p2-6p-7'  p2-l 

o^     Qv        4.1,  4.  a;2 +  8  a; +  15      x  —  1  4 

30.    Show  that 


a;2  +  7  a;  +  10      a; +  2      a;  +  2 

31.    Show  that  =-  —  1 ; ?\  =  -- 

-.1  a(a  —  l)      a 

a 
n  + 1  n  1 


32.    Show  that 


2n  +  3     2»  +  l      (2n  +  l)(2n  +  3) 


33.  Show  that  ^ ^-  =  ^  x  -%.     From  this   relation 

b     a+b      b      a+b 

find  two  numbers  such  that  their  product  is  equal  to'  their 
difference. 

34.  Simplify  the  following : 

2c? 


K^J 

c2- 

-±d2 

Na-36' 

(2) 

e 

+iX' 

'-i)(i+ 

a;2\ 

(3) 

+_»4.lYl-S 

— > 

474 


General  Review 


<*»  (!-!)•• 


,6)   g»  +  3a?  +  2x 


(5)  (2+tf 

x2  +  6  a;  -f  9 


».+  3 


x 


(7)   — +— • 
V  ;   6a2     3a 


x2  +  4x  +  3     #2  -f  4  a;  +  4 
(10)   a+^L. 


6  + 


(8) 


(9) 


x2  —  y2 


.  »+;jf 


a2  +  2a;zH-z2     cc  +  2 

(fyP  —  X*   ^        ga;2-}-a*» 

a3— a?3    '  a2+aa;-|-a;2' 


(11) 


(12) 


z  + 


ac 


6  +  c 


35.    If  2/ 


1  +  z2 


and  z  = 


1  +  x 


,  express  y  in  terms  of  x. 


36.    If  x  -f-  -  =  8,  show  that  x2  +  —  =  s2  —  2  and  a?  +  — 


a? 


=  s3-3s. 


37.    Show  that  «j 


a.* 


-2/8     »4 +  2/7     «2  +  ?/2J     »  +  2/J 


»  —  # 


1. 


38.   How  much  water  must  be  added  to  80  pounds  of  a  5  per 
cent  salt  solution  to  obtain  a  4  per  cent  solution  ?  (Yale.) 


39.    Simplify 


x  +  y- 


x  +  y- 


xy 


x  +  y. 


x3  —  y3 


xl  —  yl 


(Cornell.) 


40.  What  is  the  price  of  eggs  when  2  less  for  24  cents  raises 
the  price  2  cents  a  dozen  ?     (Yale.) 

41.  What  values  of  x  will  make  the  product  (x  —  a)  (x  —  b) 
(x  —  c)  equal  to  zero  ? 


General  Review  475 

42.  Factor  ar3  -f- 10  x2  -f  21  x  and  indicate  the  values  of  x  that 
will  make  the  expression  zero. 

43.  Simplify  the  expression : 

ftx(x  +  l)(x  +  2)+  aj(*  -  l)(a?  -  2)]+  f(*  -  l)x(x  +  1). 

44.  Divide  (a3  -  l)a3  - (.t3  +  x2  -  2)a2  +(4 x2  +  3  ar  +  2)a  -3 
(a  +  1)  by  (x  -  l)a2-(x  -  l)a  +  3. 

45.  Show    that    J  (a2  +  ?/2)  +  z2  —  J.  a?y  +  xz  —  yz    becomes 
(y  —  z)2  or  {z  —  y)2  when  —x  =  y. 

46.  By  what  transformation  can  a(x  —  b)  be  put  into  the 
form  (a  -f  b)x  —  (a  +  x)b  ? 

47.  Solve  the  following  equations  : 

(1)  2(x  -  1)=  6.  (3)  3(a  -  5)+  8  =  17. 

(2)  13(12  -  z)  =  14.  (4)  5  a  +  (7  -2x)=  11. 

(5)  8(37 -5 a)  =4(3 a- 17). 

(6)  28  +  2?/ -  16?/ -62/ -12  + 2?/ =  0. 

(7)  9x  +  22  -  2x  =  193  -  22  x  -  84. 

(8)  5x  -.3  a  =  4.5  a?  +  2. 

(9)  .9  05  -  1.5 a  =  a-  3.5. 

(10)  .25  *  +  .943  =  1.9  a?  -  6.812. 

(11)  .15  x  +  1.575  -  .875  x  =  .0625  x. 

(12)  1.111 -.Hilar  =  .3333. 

(13)  .5x  +  2-*x=Ax-ll. 

48.  Solve  the  following  equations  : 

(1)  x  +  5x-b  =2a.  (2)  3a +  2z-4b  =  5z  -  b. 

(3)  k(k  +  3  aca  +  3)  =  kx  +  3  a&A;  -  A;2  -  ac&x. 

(4)  (x-2a)2  +  (x  +  2b)2=2(x-2c)2. 
(5)x-™=p.  B±* 


(8) 


x         p 


p                 q                           in 
(7)   2.+,.  »  +',.  (9)      5x 86 


476  General  Review 

49.  Form  a  proportion  with   the  numbers  75,  18,  27,  50. 

50.  Knowing   three   terms   of  a   proportion,  how   can   the 
fourth  be  found  ? 

51.  Solve  each  of  the  following  proportions  for  x: 

(1)  ^  =  57.  (3)  3.15:  z  =  6.75:20. 
K)   10       x'  (4)  a:3f  =  4i:lli 

(2)  P2-g2.(P  +  g)2=a--6  ,6x   3^.12a  =  14c 
w    a  +  6      a2-62      p  +  g"  '       W   56 '  7c       15  6* 

52.  Form  as  many  proportions  as  possible  from  each  of  the 
following  equations : 

(1)  xy  =  vt.  (3)  (a  +  6)2  =  m2  -  n2. 

(2)  m2  =  rs.  (4)  x2  =  a2  —  b2. 

53.  Find  a  fourth  proportional  to   each   of   the   following 
sets  of  numbers : 

(1)  27,  90,  45.  (2)  p,  q,  r.  (3)  I,  1,  1. 

a    o    c 

54.  Find  a  mean  proportional  between  each  of  the  following : 
qx   a?/    ab  ,«\   2(a2  —  aft)        —  10  a 

U    6  '    ?/  '  35  6       '  7(a6  -  62) ' , 

55.  Find  a  third  proportional  to  each  of  the  following : 

(1)  8,9.  (2)     36a2&2       4(a2-q&). 

W  W    (a2  -  62)2'    b(a  +  &)2 

56.  If  -  =  - ,  prove  each  of  the  following  relations  : 

o      a 

K  }      b  d  {  ]  a-b     c-d 

(9\  a  ±  b _c  ±  d  ,,*    a-\-c_a 

1 '      a  c  U   b  +  d~b' 

57.  Find  the  values  of  x  and  y  in  each  of  the  following  : 

(1)   ?  =  ^i,whenx  +  y  =  9.        (2)  ®  =  |,  when  x  +  y  =  15. 
(3)  x:y  =  3.5  :  4,  when  x  —  y  =  2.5. 


General  Review  477 

58.  Combine  each  of  the  following  proportions  so  as  to 
eliminate  x  and  leave  the  new  proportion  in  its  simplest  form : 

^  }   d~x'  g~~f  K  ^  a-b~  (c  +  d)* 

ro\    l  —n    ®  —  -.  a        _3  c .  2c 

m     x'  p     r  14  6  lb    a 

59.  The  ratio  of  the  sun's  diameter  to  the  earth's  is  542  : 5  ; 
of  the  earth's  to  the  moon's,  11 : 3.  Find  the  ratio  of  the  di- 
ameter of  the  sun  to  that  of  the  moon. 

60.  The  age  of  a  father  to  that  of  his  son  is  as  7  to  4.  What 
is  the  age  of  each  if  the  father  is  24  years  older  than  his  son  ? 

61.  If  in  the  composition  of  powder,  the  ratio  of  niter  to 
carbon  is  31 : 9  and  of  carbon  to  sulphur  is  9 :  10,  how  much  of 
each  must  be  used  to  make  1200  pounds  of  powder  ? 

62.  Solve  by  two  methods  :  2  x  -+-  y  =  11,  3  x  —  y  =  4. 

63.  Solve  the  following  systems  of  equations : 

(1)  15a-7y  =  9,  (6)  2f  */ -  f  a;  =  90, 
9y-7x  =  13.  2|  <c  +  f  y  =  90. 

(2)  ±x  +  9y  =  106,  Sx-5y      o_2x  +  y 
8z  +  172/  =  198.  K)   '      2      "_h°-       5      ' 

(3)  3y-4:x-l  =  0,  g     x—2y==x     y 
18-3a>  =  4y.  .                4          2      3' 

(4)  5  +  4a>  =  16y,  (8)  .25a  +  3y  =  10, 
5  x  +  28  y  =  19.  4.5  x  -  4  y  =  6. 

(5)  |  a;  +  1 2/  =  34,  (9)  25.9  v  -  60.1  u  =  1, 
7  a  +  ^y  =  12.  24.1  v  -  55.9  w  =  1. 

(10)  ,2y  +  .25x  =  2(y-x), 
.8 a-  3.7 2/  =  -15.3. 

64.  The  formula  for  converting  a  temperature  of  i^7  degrees 
Fahrenheit  into  its  equivalent  temperature  C  degrees  centi- 
grade is  <7=|-(F— 32).  Express  F  in  terms  of  (7,  and  com- 
pute F  when  C  =  30°  ;  when  C  =  28°. 


478  General  Review 

65.  Find  two  numbers  such  that  their  sum  and  difference  are 
in  the  ratio  5  : 1  and  their  sum  to  their  product  in  the  ratio  5  :  8. 

66.  A  servant  is  given  $  2  to  buy  10  pounds  of  sugar  and 
4  pounds  of  cheese  and  should  have  60  cents  left.  She  makes 
a  mistake  and  buys  10  pounds  of  cheese  and  4  pounds  of  sugar 
and  lacks  24  cents.  What  is  the  price  of  cheese  and  of 
sugar  ? 

67.  Simplify  the  following : 
(1)  a2m~nam+n. 


m  (tf)X^j- 


(2)  (a  +  6)-(a  +  b): 

(3)  (p  +  qy->(p  +  gf.  (11)  ifl  x  ^JC 

a  nn  X  xP 

(4)  (a* +  &*)(a*-&").  J 

(5)  (m»-n")».  (12)^J* 

(7)  _^_  +  _# !L  (14)  [(aa;)3^4*]51'. 

m*+*      m^1       mp*  (15)  (^p  _  l)  +  (xp- 1). 

(8)  ( -  a)3(  -  a)5-  (16)  (x  +  yy-» :  (x  +  y) 


6— a 


7  —  c  \ V5  —  m\6  n  7>.    ^w+ra       ^ 


m43n 


68.  Perform  the  following  operations  and  write  the  results 
so  that  each  term  shall  have  the  integral  form  affected  by 
negative  exponents  where  necessary : 


(2)  f2x2     x     1      a     Sa,2\  --f2a     Sa' 
^  '  \a2       a  x       x2  J     \  x         x2 


69.  Write  the  following  expressions  without  using  either 
the  negative  exponent,  or  the  exponent  zero,  and  simplify  the 
results : 

(1)*.       (2)1-      (3)<rt      (4)jjL.      (6)«W.      (6)g. 


General  Review  479 

(7)  mPm-°.  (11)  (a2*"3)"2- 

(8)  (a°)5(62)-2.  (12)  (m3*"5  -  n$*-*)-*. 


y 


»  ©••  <»>  (S*E 

70.  Verify  the  identity 

a2(5-i  _  c-i)  +  fe2(c-i  _  a-i)  4.  C2(a-i  _  5-1)  _      a  +  fr  +  c 
a(6~2  -  c"2)  +  6(c"2  -  a~2)  +  c(a~2  -  b~2)       a"1  +  b~l  +  c"1 ' 

71.  Write    the    following   expressions   without    using    the 
radical  sign  or  negative  exponents : 

(1)  -Va\  (2)  (V^)6.  (3)  -V&.  (4)  Vp^. 

(5)   V{a  +  b)m+*.  (6)  ^a3-3a26  +  3a&2-&3. 

(7)  VFT.  (10)  ^/^  "£ 

<8>  ^  (11)  *L  ^ 

72.  Write  the  following  expressions  using  only  radical  signs 
and  positive  integral  exponents  : 

(1)  a*.  i      (3)  A  (5)  dl  (7)  e* 

(2)  m*.  (4)  ym.  (6)  *»«  (8)  e  *. 

73.  Simplify  the  following  radical  expressions  : 

.(1)  2Vl08a467.  (5)  ^/a2m+nb2mncm+2n. 

(2)  V7(14a-21&).  (6)      \a?_2_a  +  1 

(3)  V(n»-w)(w  +  l).  ^c2        c 

(4)  V(a2H-62)2-(2a6)2.  (7)  V98"^7VSf. 

(8)  V21+Vl3+V7+V4. 

(9)  VoVaVS.  (10)  \ 


(&+c)< 


480  General  Review 

(11)  4V32-5V50  +  3Vl8.v 

(12)  ( Va  +  3 V6) - (15  VS-2V5)  +  (4 V6  +  7  Va). 

/-.ox   Vc-hVa      Vc  —  Va        ,.,.,      /-        /-      V» 
(1^)  -—= — — -•      (14)  Vpxvgx-^- 

Va?  —  Va      V»  -f  Va  Vg 

(15)  (Vp  +  g  -  Vj>  -  g)(Vp  +  g  +  Vj>  -  g). 

(16)  Vw«-n-4-Vw  +  l.  (17)  (x  +  y)  +  iV&^jpM 

74.  Simplify  the  following  imaginary  expressions  : 

(1)  V^50+V^5+V^18+V^I-V^4  +  2V^2. 

(2)  V-3V8V-6.  VIP 

(3)  (s+V=6)(a>-V=6).  V^6 

(4)  Vl  +  V3lVl-V^l.  a^    <_(a._y)s 

(5)  Va;  —  2/V.y  —  a?,  a;  —  y  *      (x  +  yf 

(8)  (-l+V^3)3+(-l-V^3)3. 

(9)  ?? (10)  LzJ  +  l±i. 

;   7  +  2V^5  ^     '   l  +  »     l-< 

75.  Solve  the  following  equations  : 

(1)  V5a^  =  20.  (5)  VlO?/-4  =  V77TTl. 

(2)  Va7+9  =  5  VaTT3.  (6)  3Vl6a;  +  9  =  12V4al-  9. 

(3)  5  -V32/ =  4.  (7)  j  +  xl=        3       . 

(4)  V2v  +  8=V5i;  +  2.  (»  _  3)* 

(8)  * U ±—mU 

x  +  V4-x2     a;-V4-a;2       7 

(9}  Vo  a;  —  4  +  V5  —  a; _  V4a;  +  1  # 

V5  x  —  4  —  V5  —  a;      V4  a;  —  1 

76.  Solve  the  following  quadratic  equations  by  completing 
the  square  : 

(1)  z2-8a:  =  -7.  (2)  aj2  +  12  =  7». 


General  Review  481 

(3)  x(x  -  1)  =  380.  (5)  250  +  2  sfi  =  3  x>  -  W  x. 

(4)  t2  +  10 £  -  56  =  0.  (6)  6  a;2  -  |  +  4  a?  =  x  +  5  a& 

(7)  (3  a?  -  2)2=  8(aj  +  l)2-  100. 

(8)  (a; -5)2=  4. 

(9)  3a;2  +  5x-  42  =  0. 

(10)  2a;2-8  =  3a;  +  12. 

(11)  5a;(a;-2)+2=-4-a;(4a;-5). 

(12)  3(z2  +  2)2  -  54  =  3  z*  +  z(5  z  -  7). 

(13)  5(1  +  u)2  +  4  u  =  (3  -  uf  +  -4^. 

(14)a7Tl  +  2^1)=^  (16)3m2-6m=-f. 

(15)  v_3  +  ^±|  =  2v-7.         (17)  5a;2-8a;  +  3  =  0. 

(18)  (px-  2){x  +  l)  =  (a?  -  f)5  -  5. 

(19)  (1.2  -  a;)2  +  (a;  +  -8)2  =  2(6  *  -  .2)2. 
,20\  8-2 _ 8-3     s-4 _ 7 

^    '  s_i     s  +  3     s_i     4* 


77.   Solve  the  following  equations  as  quadratics  : 

(1)  ^  +  4^  =  96.  (3)  ax11  +  bx9  +  ex7  =  0. 

(2)  a/**-  16 a>*  =  512.  (4)  2ar>  -  6  -  a?i  =  0. 


(5)  IV*2  +  \x  +  81  =  J(63  -  2  a;2-  x). 

(6)  a:2  -  5  a;  +  2Va;2-5a;  +  3  =  12. 

Hint.    Add  3  to  both  members  and  treat  x2  —  5  x  +  3  as  the  unknown. 


(7)  2  a;2  -  4  a;  +  3  Va;2  -  2  x  +  6  =  15. 

(8)  a;2  +  2Va;2  +  6a;  =  24  -  6a;. 

(9)  3a;2  -  4a;  +  V3a;2  -  4a;  -  6  =  18. 

(10)  8+9V(3a;-l)(a;-2)=3a;2-7a;. 

(11)  3a;2-7  +  3V3a;2-16a;  +  21  =16  a;. 


(12)  V4  a;2  -  7  a;  -  15  -  Va;2  -  3  a;  =  Va;2  -  9. 
Hint.     The  factor  y/x  —  3  can  be  removed  from  each  expression. 


(13)  V2a?2-9a;-r-4  +  3V2a;-l=V2a;2  +  21a;-ll. 


482  General  Review 

(14)  gfi  +  x  +-  +  ~  =s4 

x     x2 

Hint.     {&  +  %  +  1\  +  x  +  ^  =  6,  or  (a  +  ^)2  +  (x  +1]-  6  =  0. 


(15)   X2  +x  +  -+*  = 

X        X2 


Hint. 


(16)  tf  + s  +  1+1  =  6f. 

(17)  12o4-56a;3  +  89a;2-56a;  +  12  =  0. 

Dividing  by  x*,  12  (x2  +  -M  -  56  (  x  +  -\  +  89  =  0.     Put  x  +  - 

^2  _i     1    _  -,2  _  9 


=  z,  then  x2  +  —  =  z2  —  2 


(18)  a^  +  ar5  -  4a;2  +  a  +  1  =  0. 

(19)  a4  +  l-3(ar5  +  a-)=2a;2. 


(20)  Vx2  +  x  + 


s/x*  —  x  2' 

78.    Solve  the  following  systems  of  quadratic  equations  : 

(1)  x2  +  I/2  =  io,  (8)  x2y  +  a*/2  =  120, 

x  —  y  =  2.  a?  +  f  =  152. 

(2)*  +  2/  =  23,  (9)a2-2/2=4,       t 

x  +  xy  =  144.  a?  +  2/=|. 

(3)  a-y=20,  (10)  l+l_  +  i  =  47, 

a*  _  xy  =  ioo.  *    xy    y 

(4)  ar2_42/2==9)  1  +  1=12. 

(5)  2 1?  +  3  «  *  20, 
Suv-  u2  =  SS. 


(11)  x  +  y  4-  x2  +  ,v2  =  If, 
2/  —  a;  +  ?/2  —  x2  =  —  1. 

(12)  ar>  -  f  =  279, 
(6)  a  +  2/  =  a,  X2  +  a.y  +  2/2  =  93. 


a?2  +  y2  =  6^2/.  1 

(13)  J-1 

1 

t/2  +  2/2  —  74. 


m  (13)  1-1=1304, 

(7)  y  -  z  =  1,  v    y  t»     w5 

2/*  =  20,  1_1=8 


General  Review  483 


(14)  aj»  +  y»=.152, 

(21)  2  *  -  5  y  =  -  18, 

x2  —  xy  -f  y2  =  19. 

3  a#  =  264. 

(15)  2a?2-3x#  +  2/2  = 

24, 

(22)  x2  -  y2  =  120, 

3*2_  5xy  +  2y2= 

=  33. 

x  +  ?/  =  20. 

(16)  *2/  +  x  =  104, 

(23)  a;2  +  y2  =  250, 

xy  -y  =  84. 

a;  -  2/  =  22. 

(17)  5*  +2y  =29, 

(24)  *2  +  y2  _  a-  +  2/  =  32, 

6  xy  =  -  105. 

2  ^2/  =  30. 

(18)  3^-2/2  =  83, 

(25)  afy*  +  3*2/  =  18, 

a;  4.  ?/  =  15. 

=  8, 

ic  +  2/  =  5. 

(19)  x  +  y  +  2Vx+y 

(26)  3*2-2/2  =  23, 

x2  +  xy  =  8. 

2*2  _  xy  =  12. 

(20)  ^  -  80, 

(27)  ^+»y  +  ^=  931, 

*=5. 

x2  +  xy  +y2  =  19. 

.V 

79.  If  rx  and  r2  are  the  roots  of  the  quadratic  equation 
x2  +  px  +  q  =  0,  show  that  rx  -f-  r2  =  —  p  and  r^  =  <?.  Also 
show  that  (x  —  rx)(x  —  r2)=  0. 

80.  Form  equations  whose  roots  are  the  following  : 

(1)  3,  5.        (2)  -  2,  7.  (3)  -  2,  -  \.     (4)  4.3,  2.5. 

(5)  3,  \.  (7)  V7,  Va_  _ 

(6)  4+V3,  4-V3,  (8)  2+V^3,  2-V-3. 
(9)  3,  i,  4,  \.           (10)  1,  - 1,  3,  -  3.        (11)  3,  0, 1. 

81.  Show  that  the  roots  of  the  quadratic  equation  ax2+  bx 

+  c  =  0are-6±V62-4ftC. 
2a 

82.  (1)  What  determines  the  nature  of  the  roots  of  a  quad- 
ratic equation  ? 

(2)  When  are  the  roots  real  ? 

(3)  When  are  they  imaginary  ? 

(4)  When  are  they  real  and  unequal? 

(5)  When  are  they  equal  ? 


484 


General  Review 


83.   Determine  whether  the  roots  of  the  following  equations 
are  real  or  imaginary : 


(1)  x*  -  x  -  12  =  0. 

(2)  x2  +  9a  +  8  =  0. 

(3)  afi  +  12  a; +  11  =  0. 

(4)  a?2 +  52  a  =  87. 

(5)  8a; +  20  =  a;2. 

(6)  16  a; -63=  a;2. 


(7)  2  a;2 -18a; +  65  =  0. 

(8)  2a;  +  5=a;2. 

(9)  x2  +  6z  +  4  =  0. 

(10)  a;2  +  8a;  +  25  =  0. 

(11)  x^  =2-  14a;. 

(12)  x^  -31  x  +  2461  =  0. 


84. 


Solve  the  following  systems  of  quadratic  equations 


0, 


(1)  xz-3y2  +  3x-l 
Sx  -y  +  13  =  0. 
(Yale.) 
(2).  5x*y*-2  =  3xy, 
x  +  5y  =  1. 

(Princeton.) 

(3)  2x  +  y  =  l, 
5X2  _  2/2  =  2. 

(Princeton.) 


(4) 


1  — a# 


?/ 


1+xy 


(Princeton.) 


(5)  a;  +  2/  =  7, 

(a?  -  1)2+  (y-2y  =  28. 
(Princeton.) 

(6)  a;2  +  a*/  =  ft 

xy  +  y2  =  ft 

(Princeton.) 

(7)  x-y-^/x-y  =  2, 
x*  -  ?/3  =  2044. 

(Yale.) 

(8)  a;2  +  2/2  =  13, 
2/2  -s  4(a?  -  2). 

(Cornell.) 

(9)  x^  +  2/2  =  xy  +  37, 
a; +  2/=  a*/ -17. 

(Columbia.) 


85.  Define  arithmetical  progression;  common  difference. 
What  is  meant  by  an  increasing  series  ?  a  decreasing  series  ? 
If  a  certain  term  and  the  common  difference  are  known,  how 
can  the  preceding  term  be  formed  ? 

86.  In  any  A.  P.  if  a  represents  the  first  term,  d  the  common 
difference,  n  the  number  of  terms,  Zthe  last  term,  and  s  the  sum 

of  the  terms,  prove  that  s  =  ^  (a  +  I)  =  ^  [2  a  +  (n  —  l)d]. 


General  Review  485 

87.  Find  Z  and  s  in  the  following  series : 

(1)  5,  8,  11,  -,  to  12  terms. 

(2)  1,  1.1,  1.2,  .-.,  to  20  terms. 

(3)  3  n,  5  n,  7  n,  •••,  to  36  terms. 

(4)  5 x,  5x  +  3 y,  5x  +  6y,  — ,  to  15  terms. 

88.  Given  (1)  a  =  7,  Z  =  —  3.5,  ti  =  36,  find  d  and  s. 

(2)  a  =  14.5,  Z  =  32,  d  =  .7,  find  n  and  s. 

(3)  a  =  2,  I  =  87,  s  =  801,  find  d  and  ». 

(4)  a  =  —  45,  w  =  31,  s  =  0,  find  d  and  Z. 

(5)  Z  =  11$,  n  =  37,  s  =  209|,  find  a  and  d. 

(6)  n  =  33,s  =  -33,d  =  -ifindaandZ. 

(7)  a  =  9,  d  =  4,  8  =  624,  find  n  and  Z. 

(8)  s  =  281 1,  d  =  f,  Z  =  15f,  find  a  and  n. 

89.  If  142  and  149  are  the  last  two  terms  of  an  A.  P.  and 
n  =  22,  find  a  and  s. 

90.  Insert  10  arithmetical  means  between  4  and  26. 

91.  Define  geometrical  progression.  What  is  the  ratio? 
If  a  certain  term  and  the  ratio  are  known,  how  can  the  preced- 
ing term  be  formed  ?  the  following  term  ? 

92.  In  any  G.  P.  if  a  represents  the  first  term,  Z  the  last 
term,  n  the  number  of  terms,  r  the  ratio,  and  s  the  sum  of  the 
terms,  prove  that 

7  „  1       j         Ir—  a      a(rn  —  1)      a(l  —  rn) 

I  =  ar"'1  and  s  = =  -^ —^  =  -*- '- . 

r  —  1  r  —  1  1  —  r 

93.  What  does  the  last  formula  of  example  92  become  when 
the  series  becomes  infinite  r  <  1  ? 

94.  Find  Z  and  s  in  the  following  series  : 

(1)  4,  8,  16,  ».,  to  7  terms. 

(2)  13,  1.3,  .13,  •-.,  to  7  terms. 

(3)  m2,  m^n,  m2n2,  — ,  to  10  terms. 

(4)  6,  -A-,        6        ,  -..,  to  9  terms. 

1  —  m    (1  —  m)2 


486  General  Review 

95.  Given  (1)  a  =  36,  I  =  J,  n  =  5,  find  r. 

(2)  Z  =  128,  r  =  2,  n  =  7,  find  a  and  s. 

(3)  a  =  3,  Z  =  192  V2,  r  =  V2,  find  s  and  n. 

(4)  a  =  10,  Z  =  ^,  «  =  19ft,  find  r  and  ». 

96.  Find  the  two  unknowns  in  each  of  the  following : 

(1)  a  =  3,  r  =  3,  s  =  29,523.       (3)  I  =  1296,  r  =  6,  s  =  1555. 

(2)  r  =  2,  w  =  7,  s  =  635.  (4)  a  =  18,  n  =  3,  s  =  1026. 

97.  Find  the  sum  of  the  series  9,  —  3,  1,  —  i,  —,  to  in- 
finity. 

98.  Find  the  sum  to  infinity  of  the  series,  1,  —  J,  J,  —  \ 
—.     Also  find  the  sum  of  the  positive  terms.  (Yale.) 

99.  Expand  each  of  the  following  by  the  binomial  formula : 
(1)  (b  +  x)\      (2)(a-xy.      (3)  (2a>+3a)«.      (4)  ( V^  +  xf. 

100.  Find  the  6th  term  in  the  expansion  of  (3  +  2  #2)9  and 
the  7th  term  in  the  expansion  of  (J  a  —  x)17. 

101.  Expand  (2x*  —  y3)s.  (University  of  Michigan.) 

102.  Raise  98  to  the  5th  power  by  the  Binomial  Theorem. 
(Write  98  =  100  -  2.)  (Yale.) 

103.  Find  the  first  three  terms  of  (1  +  2  a;)8  by  the  Binomial 
Theorem.  (Sheffield  Scientific  School.) 

104.  Find  the  coefficient  of  a?  and  x4  in  the  expansion  of 
(1+2  x)s,  using  the  Binomial  Theorem. 

(Sheffield  Scientific  School.) 

105.  Expand  the  expression  (x*  —  x~*y  and  write  the  result 
in  a  form  free  from  negative  exponents.  (Harvard.) 

106.  Define  logarithm  ;  base.     In  the  equation  a*  =  N,  what 
is  x  ?  what  is  a  ? 

107.  What  are  the  logarithms  of  3,  81,  243,  729  in  a  system 
of  which  the  base  is  3  ? 

108.  What  are  the  logarithms  of  J,  ^7,  -fa,  ^  in  a  system 
of  which  the  base  is  3  ? 


General  Review  487 

109.  Given  log10  3  =  .477  ;  find  log3 10,  log3 .1,  log3 .01. 

(Sheffield  Scientific  School.) 

110.  Compute  the  value  of  x  from  the  equation 

=  (39.71)3Vl3:i6 
X  (46.71)4 

using  logarithms.  (Sheffield  Scientific  School.) 

111.  Perform  the  following  operations,  using  logarithms  : 

(1)  95x34.  (3)  ^76245.  (5)  154-235. 

(2)  ^4158.  (4)  ^2.  (6)   **g». 

112.  Solve  the  following  equations : 

(1)  2*  =  1024.  (3)  12-  =  20,737. 

(2)  10,000-  =  10.  (4)  31**  =  4. 

113.  Write  the  roots  of  (x2-\-2  x)(x2-2  x-3)(x2-x  +  l)  =  0. 

(Sheffield  Scientific  School.) 

114.  Solve  the  equation  x2  —  1.6  x  —  .23  =  0,  obtaining  the 
values  of  the  roots  correct  to  3  significant  figures.     (Harvard.) 

115.  The  distance  s  that  a  body  falls  from  rest  in  t  seconds 
is  given  by  the  formula  s  =  16 t2.  A  man  drops  a  stone  into  a 
well  and  hears  the  splash  after  3  seconds.  If  the  velocity  of 
sound  in  air  is  1086  feet  a  second,  what  is  the  depth  of  the 
well  ?     (Yale.) 

116.  A  man  spent  $539  for  sheep.  He  kept  14  of  the 
flock  that  he  bought  and  sold  the  remainder  at  an  advance  of 
$2  per  head,  gaming  $28  by  the  transaction.  How  many 
sheep  did  he  buy  and  what  was  the  cost  of  each  ?     (Yale.) 

117.  Solve  by  factoring  a?3  +  30  x  =  11  x2. 

(Colorado  School  of  Mines.) 

118.  Solve  the  equation  .03  x2  -  2.23  x  +  1.1075  =  0. 

(Colorado  School  of  Mines.) 


488  General  Review 

119.  How  many  pairs  of  numbers  will  satisfy  simultaneously 
the  two  equations  3  x  +  2  y  =  7  and  x  +  y  =  3  ?  Show  by  means 
of  a  graph  that  your  answer  is  correct.  What  is  meant  by 
eliminating  x  in  the  above  equations  by  substitution?  by 
subtraction?     (Colorado  School  of  Mines.) 

120.  An  automobile  went  80  miles  and  back  in  9  hours.  The 
rate  of  speed  returning  was  4  miles  per  hour  faster  than  the 
rate  going.     Find  the  rate  each  way.     (Cornell.) 

121.  A  goldsmith  has  two  alloys  of  gold,  the  first  being  f 
pure  gold,  the  second  -^  pure  gold.  How  much  of  each  must 
he  take  to  produce  100  ounces  of  an  alloy  which  shall  be  -§  pure 
gold  ?     (Harvard.) 

122.  A  man  walked  to  a  railway  station  at  the  rate  of  4 
miles  an  hour  and  traveled  by  train  at  the  rate  of  30  miles  an 
hour,  reaching  his  destination  in  20  hours.  If  he  had  walked 
3  miles  an  hour  and  ridden  35  miles  an  hour,  he  would  have 
made  the  journey  in  18  hours.  What  was  the  total  distance 
traveled  ?     (Mass.  Institute  of  Technology.) 

123.  A  page  is  to  have  a  margin  of  1  inch,  and  is  to  contain 
35  square  inches  of  printing.  How  large  must  the  page  be,  if 
the  length  is  to  exceed  the  width  by  2  inches  ? 

(Mount  Holyoke  College.) 

124.  Factor  the  following  expressions  : 

(a)  a* -6* 

(b)  xY*2  —  ®2z  —  y2%  4- 1. 

(c)  16(0?  +  y)A  -  (2  x  -  y)\     (Mount  Holyoke  College.) 

125.  If  four  quantities  are  in  proportion  and  the  second  is  a 
mean  proportional  between  the  third  and  the  fourth,  prove 
that  the  third  will  be  a  mean  proportional  between  the  first 
and  the  second.     (Princeton.) 

126.  Solve  x2  +  y2  —  xy  =  7. 

x  +  y  =  4.     (Smith  College.) 


General  Review  489 

127.  The  diagonal  of  a  rectangle  is  13  feet  long.  If  each 
side  were  longer  by  2  feet,  the  area  would  be  increased  by 
38  square  feet.     Find  the  lengths  of  the  sides.     (Smith  College.) 

128.  A  field  could  be  made  into  a  square  by  diminishing  the 
length  by  10  feet  and  increasing  the  breadth  by  5  feet,  but  the 
area  would  then  be  diminished  by  210  square  feet.  Find  the 
length  and  the  breadth  of  the  field.     (Vassar  College.) 

129.  Simplify  — — ~*~         ,  and  compute   the  value  of  the 

V2-V12 
fraction  to  two  decimal  places.     (Yale.) 

130.  In  going  7500  yards  a  front  wheel  of  a  wagon  makes 
1000  more  revolutions  than  a  rear  one.  If  the  wheels  were 
each  1  yard  greater  in  circumference,  a  front  wheel  would  make 
625  more  revolutions  than  a  rear  one.  Find  the  circumference 
of  each.     (Yale.) 

131.  In  the  expansion  of  (2  x  —  3  a;-1)8,  find  the  term  that 
does  not  contain  x.     (Princeton.) 


COLLEGE  ENTRANCE  EXAMINATIONS 

UNIVERSITY   OF    CALIFORNIA 

Elementary  Algebra 

1.  At  a  football  game  there  were  sixteen  thousand  persons.  The  num- 
ber of  women  was  six  times  the  number  of  children  and  the  number  of 
men  was  three  thousand  less  than  twice  the  number  of  women.  How 
many  men,  women,  and  children  were  there  ? 

2.  Solve  :  (a)  2x  +  3  =  0. 

(&)  3n  +  2(n  +  4)=4n  +  14. 

3.  Factor :  (a)  a4  -  7  z2  +  12. 

(6)   &s_27. 

(c)   (3s-  l)2-(x2  +  4y2-4xy). 

4.  The  difference  between  two  numbers  is  14  and  their  product  is  176. 
Find  the  numbers. 

5.  Solve  for  x  :    (a)   ?  =  — 2L.  • 

v  J   b     c-x 

(6)  x(x2  -  4)  (z2  -  9)  =  0. 

6.  («)  Describe  the  method  of  locating  a  point  on  squared  paper. 
(6)  Construct  a  graph  of  (a)  2  x  =  5  y  +  10,  (&)  x  =  5. 

7.  Solve  graphically  :      j*  +  2y=4' 
2x  +  y=-l. 


CORNELL   UNIVERSITY 

Elementary  Algebra 

1.  Multiply  1  +  2  x  —  x2  —  \  xs  by  itself,  and  then  find  the  value  of  the 
result  if  1  -  2  x  =  3. 

2.  What  is  the  value  of  x*  +  y3  if  x  +  y  =  4,  and  2x2-f2  y2  =  17  ? 

490 


College  Entrance  Examinations  491 

3.    (a)  Add  -,    ,        x  ~ — ,  and  — — ,  and  express  the  result 

v   J  x     1  -x     (x  +  l)2  x  +  1 

as  a  fraction  in  its  lowest  terms. 

Vx  —  4Vx—  2 


(6)  Rationalize  the  denominator  of  ,- 


2Vx  +  3Vx-2 

4.  Find  a  root  of  x2  —  x  —  1  =0,  and  verify  correctness  of  the  result. 

5.  Solve  :  2x  +  4y  +  5z  =  19, 

-3x  +  5y  +  7z  =  8, 
Sx-Sy  +  50  =  23. 

6.  A  takes  three  hours  longer  than  B  to  walk  30  miles;  but  if  A 
doubles  his  pace  he  takes  two  hours  less  than  B.  Find  the  rate  at  which 
A  and  B  each  walk. 

7.  Find  the  time  between  three  and  four  o'clock  when  the  minute  and 
hour  hands  are  opposite  each  other. 

Intermediate  Algebra 

1.  Solve  for  x  and  check  results  : 


x2  -  1     x-1     x+1 

2.  Solve  and  check  :  x  +  y  +  2  y/x  +  y  —  1  =  25, 

x  —  y  +  3Vx—  y  +  1  =  9. 

3.  For  what  values  of  m  will  the  roots  of  2  m2  +  x2  —  2  mx  +  4  x  —  6  m 
+  4  =  0  be  real  and  distinct  ? 

4.  Evaluate  (81)^+ (  -  27)^ 

3(9)"*  +  (27)"1 

5.  Find  the  greatest  common  divisor  of  x4  +  x3  —  x2  —  x  and  x4  +  4x3 
+  3  x2  —  4  x  —  4.      Also  find  the  least  common  multiple. 

6.  What  is  the  sum  of  1  +  3  +  5  +  •••  +  (2  n  —  1),  n  being  a  positive 
integer  ?  What  is  the  least  odd  integer  such  that  the  sum  of  all  the  posi- 
tive odd  integers  up  to  and  including  it  will  exceed  45,370  ? 

7.  From  a  thread  whose  length  is  equal  to  the  perimeter  of  a  square, 
1  yd.  is  cut  off.  The  remainder  equals  the  perimeter  of  a  square  whose 
area  is  |  that  of  the  first.     What  was  the  original  length  of  the  thread  ? 


492  College  Entrance  Examinations 


PRINCETON   UNIVERSITY 
Algebra  A I 
1.    Simplify  — ■*-(«-  8). 


2  +  *i±_3 


2.  Simplify  {x~%{xy-2)~^(x-^)Y ',    3  V}  +  V40  +  V|  -  v'J, 

3.  Factor  x2  -  4  ax  -  4  b2  +  8  a&  ; 

a2  +  cd  -  a&  —  bd  +  ac  +  ad ; 
(x  +  l)(6x2-x)-15(x  +  1). 

4.  Find  the  H.  C.  F.  of 

x4  -  2  «3  -  Sx2  -  2  x  -  4  and  x4  -  x3  -  7  x2  -  2x  +  4. 

5.  Solve  Z  +  2/+    (si, 

2x  +  ?/  +  3£=4, 
3x  +  ?/  +  7£  =  13. 

6.  A  gave  B  as  much  money  as  B  had  ;  then  B  gave  A  as  much  money 
as  A  had  left ;  finally  A  gave  B  half  as  much  as  B  then  had  left.  A  ends 
with  $  4  and  B  with  $  36  ;  how  much  had  each  originally  ? 


1.   Solve  (a)  x-^  =  b 


Algebra  A  II 

a  _  b 
b     a 


(&) 


Vx+  1       5 


x  +  Vx  +  1      n 

2.  Solve  the  following  equations,  pairing  the  corresponding  values  of 
x  and  y  and  testing  one  solution  in  each  case  : 

(a)       *+2>  =  4'  (b)     2(x2  +  2/2)+x  +  t/=  11, 

x     j/_5.  xy  =  l. 

y     x     2 

3.  Show  that  the  series  whose  terms  are  the  reciprocals  of  the  terms  of 
a  G.  P.  is  a  G.  P. 

How  many  terms  of  the  progression  f,  |f,  -^  •••  must  be  taken  to  make 
the  sum  36£  ? 

4.  A  earned  $6,  and  B,  who  worked  4  days  more  than  A,  earned  $  14. 
Had  their  wages  per  day  been  interchanged  they  would  together  have 
earned  $  19.     How  many  days  did  each  work  ? 


College  Entrance  Examinations  493 

YALE   UNIVERSITY— SHEFFIELD   SCIENTIFIC   SCHOOL 
Elementary  Algebra 

(Omit  one  question  in  1-3  and  one  in  7-9) 

1.   Solve  3_      1  3  5 


x  +  2      4z-6     2s2  +  3-6 


2.  Solve  and  verify   V5  -  2  x  +  Vl5  —  3  x  =  V26  —  6  a. 

3.  Draw  the  graphs  of  the  equations  : 

y  =  x-l, 
y  =  x2  —  4  x  +  5. 
Solve  them  simultaneously  and  explain  the  relation  between  the  graphs 
and  the  solutions. 


4.    Simplify  (a)   ^-±-^ 2(1 ^—\ 

*    *   K  J   x*  +  4x  +  3        V         1  +  aJ 


27  +  xs 


^  2  n(m  —  n) 

,-.  s    m  +  n 

(6)  i^  +  n'      ! 

mn  +  w2 

6.    Simplify  (a)  -JI+-JS"-       (6)   -  J  m~*n       +  m-^' 
8         lb  mhm-n)» 

6.  Write  the  first  three  and  last  three  terms  of  the  expansion  of 

(a-1  +  2  a^)8  and  simplify  the  result. 

7.  A  and  B  started  in  business  at  the  same  time.  The  first  year  A  lost 
.$>  5000,  but  during  the  second  year  gained  25  %  on  the  amount  left  at  the 
end  of  the  first  year.  B  started  with  f  as  much  money  as  A  and  gained 
20%  the  first  year,  hut  lost  $2050  the  second  year.  He  then  had  the 
same  amount  as  A.     How  much  had  each  at  first  ? 

8.  A  13-foot  ladder  leaning  against  a  building  lacks  3  feet  of  reaching  a 
window,  while  a  17-foot  ladder  with  its  base  placed  3  feet  farther  from  the 
wall  just  reaches  it.  How  high  is  the  window  from  the  ground  and  how 
far  was  the  bottom  of  the  first  ladder  from  the  wall  ? 

9.  Divide  S  700  between  A,  B,  C,  and  D,  so  that  their  shares  may  be  in 
geometrical  progression  and  the  sum  of  A's  and  B's  shares  equal  to  $  262. 


APPENDIX 

REMAINDER   THEOREM,    FACTOR   THEOREM, 
AND    SYNTHETIC    DIVISION 

647.   Remainder  Theorem. 
1.    Divider2—  5x+8byx  —  a.      2.    Divide  x2  —  5x+8  by  x  —  2. 


x*-5x  +  8 

£  —  a 

X2-5z  +  8 

z2-2sc 
-3* +8 
-3z  +  6 

z-2 

x2  —  ax 

x  +  (a  - 

-5) 

x-3 

(a  -  5)z  +  8 
(a  -  5)x  -  a2 

+  5a 

a2  -  5 a  +  8  =  Rem.  2  =  22— 5-2  +  8  =  Rem. 

Note  that  the  remainder  in  the  first  division  is  the  same  as  the  dividend 
except  that  a  has  replaced  x.  The  remainder  when  dividing  by  x  —  2  is 
the  result  of  substituting  2  for  x  in  the  dividend. 

648.  If  a  rational  and  integral  expression  in  x  is  divided  by  x  —  a 
until  the  remainder  does  not  contain  x,  the  remainder  is  the  expression 
obtained  by  substituting  a  for  x  in  the  dividend. 

Proof.  Call  the  dividend  Dx,  and  let  J)a  be  the  result  of  putting  a  for 
x  in  the  dividend.  Call  the  quotient  Qx,  and  let  Qa  be  the  result  of  sub- 
stituting a  for  x  in  the  quotient  and  let  the  remainder  be  B. 

Dx=(x  —  a) Qx  +  B.     (Dividend  =  divisor  x  quotient  -f  remainder.) 

This  equation  is  an  identity  ;  that  is,  it  is  true  for  all  values  of  x.  Sub- 
stitute a  for  x  in  both  members. 

Then  Da  =  (a-  a)  Qa  +  B. 

.-.  Da  =  0-  Qa  +  B  or  Da  =  B.  q.e.d. 

The  student  should  note  that  R  is  not  changed  when  a  is  substituted 
for  cc,  since  B  does  not  contain  x. 

404 


Appendix  495 

Examples 

1.  Find  the  remainder  when  sc3  —  x  +  7  is  divided  by  x  —  3. 

Solution.     Substitute  3  for  x. 

33  _  3  +  7  —  31,  the  remainder. 

2.  Find  the  remainder  when  x3  —  5  x2  +  2  is  divided  by  #  4-  2. 

Solution.  x  +  2  =  x  -(  —  2). 

Substitute  —  2  for  x. 

(_  2)3  —  5(-  2)2  +  2  =—  8  —  20  +  2  =—  26,  the  remainder. 

EXERCISE 

649.  Find  the  remainder  when  each  expression  is  divided  by 
the  binomial  opposite  it: 

8.  x3  —  2  a +  3,  x  +  1. 

9.  x*  +  l,  a>  +  l. 

10.  ic5  +  1,  x  +  1. 

11.  #7  +  l,  aj  +  1- 

5.  3^-17x-30,  z-3.         12.   a3-l,  a?— 1. 

6.  y3  —  5  ?/  +  4,  ?/  -  3.  13.    a;3  +  #2  +  x  +  1,    a  +  2. 

7.  a8  —  2s8 +  3,  z-1.         14.   a?  +  a;2  +  a>  +  l,    a  +  l. 

650.  Factor  Theorem.  The  student  will  observe  that  in 
some  of  the  examples  in  §  649  the  remainder  was  0,  and,  there- 
fore, the  divisor  was  a  factor  of  the  dividend. 

651.  If  a  rational  integral  expression  containing  x  vanishes  (becomes 
equal  to  0)  when  a  is  put  for  x,  then  x  —  a  is  a  factor  of  the  expression. 

Proof.  If  the  result  of  substituting  a  for  x  is  0,  the  remainder  when 
the  expression  is  divided  by  x  —  a  is  0,  and  the  divisor  is  a  factor  of  the 
dividend.      (§648.) 

Examples 

1.    Show  that  x  —  1  is  a  factor  of  x?  —  2  x  +  1. 

Solution.     Substituting  1  for  x  gives 
18_  2. 1  +  1  =  0;  .'.  x  —  1  is  a  factor. 

By  dividing  x3  —  2  x  +  1  by  x  —  1  the  other  factor  is  found  to  be 
x*  +  x-l.     .-.  x*-2x+  1  =  (x-l)(se2  +  x-  1). 


1. 

a? -3a; +  8, 

aj-1. 

2. 

a**_2ic  +  3, 

a -2. 

3. 

2  «2  -  5  a  +  4, 

ic  —  5. 

4. 

#-2  x  4-1, 

x-1. 

496  Appendix 

2.   Factor  a?  -  5  x  -  12. 

The  number  to  substitute  for  x  must  be  found  by  trial.    Try  different 
factors  of  12.  3a_5.3_12  =  o. 

.\x  —  3  is  a  factor. 
z*-5x-  12  =  (x-3)(x2  +  3x  +  4). 


EXERCISE 

652.  Factor  by  the  factor  theorem : 

1.  x*  -7x  +  6.         4.   a;2 -5a; -6.         7.   3a?  +  a;2 -28. 

2.  a?  —  8.  5.   ar>-2a;2-9.        8.   2a?  +  a;2—  a;-  2. 

3.  a;2  -5a;  +  6.         6.   a?  -  2a;  -  56.       9.  x*  +  2x  +  l. 

Solution.  All  terms  are  positive,  so  no  positive  number  need  be 
tried.    (Why?)       (_1)4  +  a(_  1)+1  =  ft 

.  \  x—  ( —  1)  or  x  -f  1  is  a  factor. 
a4  +  2x  +  l=(x+l)(     ?    ). 

10.  x3  +  3a;-f-14.  13.   2a?  +  3a!+5. 

11.  a?-6a;2  +  lla;-  6.  14.  5  xi  -  21  a;  -  38. 

12.  a? -6ar>  + 13a; -10.  15.  7  a? +  9  a; +  16. 

653.  Synthetic  Division.  Division  of  polynomials  containing 
x  by  divisors  of  the  form  x  —  a  can  be  performed  very  ex- 
peditiously by  the  method  known  as  synthetic  division. 


xs  _  6  x2  +  11  x  +  2 

x-2 

-  4  x2  +  ]>£ 

z^Ar&+    8x 

x2  -  4  x  +  3 

3x+# 
3^-6 

8 

The  coefficients  of  the  first  term  and  of  each  partial  remainder,  except 
the  last,  are  the  coefficients  of  the  terms  of  the  quotient.  The  terms 
crossed  off  could  as  well  as  not  be  entirely  omitted  from  the  work.    The 


Appendix  497 

se's  could  all  be  omitted,  since  the  orderly  arrangement  of  the  work  would 
enable  us  to  replace  each  x  with  the  proper  exponent.  The  subtractions 
that  we  made  in  the  original  division,  —  2x2  from  -6a;2;  Sx  from  11  x; 
—  6  from  2,  can  be  changed  to  additions  by  using  2  instead  of  —  2  as  a 
multiplier. 

The  work  is  thus  reduced  to  the  following : 

1-6  +  11  +  2 

2-    8  +  6 


1  —  4  +    3 1+  8,    remainder. 

1.  Write  the  coefficients  of  the  dividend  in  order  with  their  signs  and 
write  the  second  term  of  the  divisor  with  its  sign  changed  to  the  right. 

2.  Bring  down  the  coefficient  of  the  highest  degree  term,  1,  as  the 
coefficient  of  the  first  term  of  the  quotient. 

3.  Multiply  this  number  by  2  and  write  the  product,  2,  under  the 
second  term  and  add.  This  gives  —  4,  the  coefficient  of  the  second  term 
of  the  quotient. 

4.  Multiply  —  4  by  2  and  write  the  product  under  11  and  add. 
6.  Repeat  this  process  to  the  end  of  the  polynomial. 

6.  The  first  three  numbers  on  the  last  line  are  the  coefficients  of  the 
quotient  x2  —  4  x  +  3,  and  the  last  number  is  the  remainder. 

If  any  power  of  x  from  the  highest  down  to  the  absolute  term  is  miss- 
ing, a  zero  coefficient  must  be  supplied  in  its  place  in  the  detached  coeffi- 
cients. 

Examples 

1.   Divide  £c4  +  aJ2+-8byz-l. 

Division :  1+0+1+0  +  8 

1+1+2+2 


1+1  +  2  +  2 1+  10,  remainder. 
The  quotient  isxz  +  x2  +  2x  +  2  and  the  remainder  is  10. 

2.   Divide  x3  +  6  x2  -16  by  x  +  2. 

Division  :  1+6  +  0—  16 1-2 

-2-8+161 
1  +  4-8       10 

The  quotient  is  x2  +  4  x  -  8  and  the  division  is  exact 


498  Appendix 

EXERCISE 

654.    Divide  the  following  by  synthetic  division  : 

1.  (a;2_9x  +  20)--O-l).         7.    (3x?-5x  +  2)  +  (x-2). 

2.  (;xZ-9x  +  20)  +  (x-2).         8.    (x3  4-  2x  +  l)-s-(a>  +  1). 

3.  (a* -9  a?  +  20) -*-(»- 4).        9.   (^  +  1)^  +  1). 

4.  (x2-9x  +  20)  +  (x-5).       10.    (x3-1)h-(«-1). 

5.  («3-9ic  +  20)--(x-2).       11.    (a*-l)  +  (x  +  l). 

6.  (2#-9«  +  3)-i-(aj-3). 

12.    Factor  a4  -  5a3  +  3 x2  +  15 a  -  18. 

Solution.     Substituting  2  for  x  makes  the  expression  vanish.    There- 
fore x  —  2  is  a  factor.     Divide  by  X  —  2. 

1_5  +3  +  15_  1812 

2-6-    6  +  18| 
1_3_3+    9       |_o 

The  quotient  x3  —  3x2— 3x  +  9  vanishes  for  x  =  3.     Divide  by  x  —  3. 

1  -3-3  +  913 
3-4-0  — p| 

l  +  0-3    [0 
The  quotient  is  x2  —  3. 
.-.  x4  -  5x3  +  3x2  +  16x  _  18  =  (X  _  2)(«  -  3) (x2  -  3). 

Factor  the  following  : 

13.  o#  — 10.S  +  &  17.  a?  +  a?2  +  a;  —  3. 

14.  2a3+-5#2-4.  18.  a3+-a  +  2. 

15.  ^-#  +  24.  19.  a3-  3z  -322. 

16.  a?— 13  a* +  49  a>  — 45.  20.  a3  +  5a; +  150. 


INDEX 

[The  numbers  refer  to  pages.] 


Abscissa,  245. 
Absolute  terra,  290. 
Absolute  value,  20. 
Addition,  32-49. 

of  fractions,  179. 

of  imaginary  numbers,  369. 

of  like  monomials,  35. 

of  polynomials,  42. 

of  radicals,  341. 

of  signed  numbers,  18,  20. 

of  unlike  monomials,  40. 
Addition  and  subtraction,  elimination 

by,  255. 
Algebraic  expression,  32,  93. 
Algebraic  fraction,  1(57. 
Algebraic  improper  fraction,  174. 
Algebraic  number  scale,  17. 
Algebraic  numbers,  18. 
Algebraic  signs  in  fractions,  171. 
Algebraic  sum,  22. 
Alternation,  proportion  by,  226. 
Antecedent,  218,  219. 
Antilogarithms,  464. 
Appendix,  494-498. 
Area  of  circle,  rectangle,  triangle,  1. 
Arithmetical  mean ,  429. 
Arithmetical  number,  square  root  of, 

283. 
Arithmetical  progression,  427. 
Arranging  terras  of  a  polynomial,  39. 
Ascending  powers,  39. 
Axis,  244. 


Base,  33. 

Binomial,  32. 

Binomial  coefficients  and  exponents, 

444. 
Binomial  formula,  442-447. 
Binomial  quadratic  surd,  356. 
Braces,  58. 
Brackets,  58. 
Briggs's  logarithms,  458. 


Cancellation,  168. 

Characteristic,  458,  459. 

Check,  8,  44. 

Circumference  of  Circle,  1, 

Clearing  of  fractions,  197. 

Coefficient,  33,  444. 

Collecting  terms,  41. 

College  Entrance  Examinations,  490. 

Cologarithra,  465. 

Common  denominator,  177. 

Common  difference,  427. 

Common  factor,  162,  163. 

Common  logarithm,  458. 

Common  multiple,  165. 

Complete  quadratic,  290,  294,  374. 

Completing  the  square,  294,  296. 

Complex  fraction,  194. 

Complex  number,  368. 

Composition,  proportion  by,  227. 

Composition  and  division,  proportion 

by,  230. 
Compound  interest,  466. 
Conditional  equation,  104. 
Conjugate  quadratic  surd,  350. 
Consequent,  218,  219. 
Constant  of  variation,  448. 
Coordinates,  244,  245. 
Cube,  5. 
Cube  root,  5. 

Decimal,  repeating,  437. 
Deduction,  symbols  of,  5. 
Degree,  162. 

Degree  of  an  equation,  400. 
Denominator,  167. 

lowest  common,  177. 

rationalizing,  352. 
Descending  powers,  39. 
Difference,  in  subtraction,  22. 

common,  427. 
Direct  variation,  448. 
Discriminant,  386. 
Dissimilar  terms,  35. 


499 


500 


Index 


Dividend,  92,  167. 
Division,  92-103. 

defined,  27,  92. 

of  fractions,  190. 

of  imaginary  numbers,  369. 

of  monomials,  94. 

of  polynomials,  96,  99. 

of  radicals,  344,  350. 

of  signed  numbers,  27. 

proportion  by,  228. 

synthetic,  496. 
Divisor,  92,  167. 

Elimination,  by  addition  and  subtrac 
tion,  255,  412. 
by  substitution,  259,  412. 
defined,  252. 
Equality,  symbol  of,  2. 
Equation,  complete  quadratic,  290. 
conditional,  104. 
defined,  7,  104. 
degree  of,  400. 

formed  with  given  roots,  387. 
identical,  104. 
incomplete  quadratic,  290. 
indeterminate,  250. 
integral,  107. 
linear,  107. 
members  of,  7. 
principles  used  in  solving,  8. 
radical,  362. 
simple,  104,  107. 
type  form  ax=  b,  108. 
Equations,  containing  fractions,  197- 
217. 
determinate  system,  250. 
graphical  solutions,  253-254, 418-426. 
homogeneous,  402. 
independent,  251. 
linear  simultaneous,  250,  251. 
literal,  201. 
quadratic  form,  388. 
quadratic  simultaneous,  305. 
symmetrical,  407. 
Evolution  of  radicals,  355. 
Exponent,  defined,  5,  33,  322,  454. 
fractional,  316. 
negative,  318. 
zero,  318. 
Exponents,  315-330.  / 


Exponents,  definitions  of,  322. 

laws  of,  67,  93,  315,  322. 
Expression,  integral  algebraic,  93. 

mixed,  174. 

radical,  332. 
Extraneous  root,  363. 
Extremes,  219. 

Factor,  common,  162. 
defined,  5,  25,  132. 
highest  common,  163. 
rationalizing,  349. 
Factor  theorem,  495. 
Factoring,  132-160. 
cases  of,  133-150. 
solution  of  equations  by,  156. 
square  root  by,  280. 
summary  of,  152. 
Formula,  binomial,  442. 
defined,  1. 

for    solving    quadratic    equations, 
381. 
Formulas,  rules  and,  123,  124. 
Fourth  proportional,  220. 
Fraction,  algebraic,  167. 
improper,  174. 
lowest  terms  of,  168. 
reduction  of,  168. 
terms  of,  167. 
Fractional  equations,  197,  214. 
Fractional  exponents,  316. 
Fractions,  167-196. 
addition  of,  179. 
clearing  of,  197. 
complex,  194. 
division  of,  190. 
multiplication  of,  186. 
reduction  of,  168. 
subtraction  of,  179. 
Function,  defined,  245. 
graph  of,  246. 

General  review,  469-489. 
Geometrical  mean,  435. 
Geometrical  progression,  434. 
Graph  of  function,  246. 
Graphical  solution  of  equations,  253, 

254,  418-426. 
Graphs,  237-249,  253,  254,  418-426. 


Index 


501 


Highest  common  factor,  161-165. 

defined,  163. 
Homogeneous  equations,  402. 

Identity,  104. 

Imaginary  numbers,  367-372. 

addition  and  subtraction  of,  369. 

denned,  331,367. 

division  of,  369. 

multiplication  of,  369. 

operations  with,  368. 

reduction  of,  369. 

unit,  powers  of,  368. 
Improper  fraction,  174. 
Incomplete  quadratic,  290,  291. 
Independent  equations,  251. 
Index  of  radicals,  332. 
Infinite  geometrical  series,  437-438. 
Inserting  parentheses,  62. 
Integral  algebraic  expression,  93. 
Integral  equation,  107. 
Integral  polynomial,  161,  162. 
Integral  term,  161. 
Interest,  compound,  466. 
Introduction,  1-14. 
Inverse  variation,  451. 
Inversion,  proportion  by,  226. 
Involution  of  radicals,  355. 
Irrational  equation,  362. 
Irrational  numbers,  331. 

Joint  variation,  452. 

Last  term,  428,  434. 
Laws  of  exponents,  67,  93,  315-322. 
Like  terms,  or  monomials,   addition 
of,  35. 

defined,  35. 

subtraction  of,  50. 
Linear  equations,  107,  250-278. 
Literal  equations,  201. 
Logarithms,  454-468. 

common  or  Briggs,  458. 

laws  of,  456-457. 

table  of,  461-463. 
Lowest  common  denominator,  177. 
Lowest  common  multiple,  165,  166. 

Mantissa,  458,  459. 
Mean,  arithmetical,  429. 


Mean,  geometrical,  435. 

Mean  proportional,  220. 

Means,  219. 

Members  of  equations,  7. 

Minuend,  22,  23. 

Mixed  expression,  174. 

Monomial,  addition  of,  35,  40. 

defined,  32. 

division  of,  94. 

multiplication  of,  69. 
Multiple,  lowest  common,  165. 
Multiplication,  67-91. 

of  fractions,  186. 

of  imaginary  numbers,  369. 

of  polynomials,  71,  74. 

of  radicals,  344. 

of  signed  numbers,  25. 

type  forms  of,  78-87. 

Nature  of  roots,  385. 
Negative  exponents,  318. 
Negative  numbers,  15.       v 
Number  scale,  17,  18. 
Numbers,  algebraic,  18. 

complex,  368. 

literal,  1. 

positive  and  negative,  15,  16. 

prime,  132. 

rational  and  irrational,  331. 

real  and  imaginary,  331,  367° 

signed,  18. 

symbols  representing,  1. 

unknown,  7. 
Numerator,  167. 
Numerical  coefficient,  33. 

Operation,  order  of,  28. 

symbols  representing,  2. 
Order  of  radicals,  332. 
Ordinate,  245. 

P-form  of  quadratic,  295. 
Parenthesis,  explained,  57. 

inserting  in,  62. 

removal  of,  59. 
Polynomials,  addition  of,  42. 

defined,  32. 

degree  of,  162. 

division  of,  96,  99. 

multiplication  of,  71,  74. 


502 


Index 


Polynomials,    rational  and   integral, 
161. 

square  root  of,  280. 

subtraction  of,  54. 
Positive  and  negative  numbers,  15-31. 
Power,  defined,  33,  279. 

of  imaginary  unit,  368. 
Powers,  ascending,  39. 

descending,  39. 

of  monomials,  355. 
Prime  factor,  132. 
Prime  number,  132. 
Principal  root,  332. 
Principles  used  in  solving  equations,  8. 
Problems,  hints  on  the  solution  of,  116. 
Product,  5,  25,  132. 
Progression,  arithmetical,  427-433. 

geometrical,  434-411. 
Proportion,  terms  of,  218,  219. 
Proportional,  fourth,  mean,  third,  220. 
Proportions,  properties  of,  222. 

Quadrants,  244. 

Quadratic,  complete,  290,  294,  374. 

equations  in  the  form  of,  388. 

incomplete,  290,  291. 

p-form,  295. 

relation  between   roots  and  coeffi- 
cients, 384. 

solution  by  completing  the  square, 
296,  374. 

solution  by  factoring,  156,  377. 

solution  by  formula,  380. 

surd,  332,  350. 

theory  of,  384. 
Quadratic    equations,    290-304,    373- 
399. 

formula  for  solving,  381. 

simultaneous,  305-314,  400-417. 

theory  of,  384. 
Quadratic  trinomial,  146. 
Quotient,  27,  92. 

Radical  equations,  362-365. 
Radical  expressions,  332. 
Radical  index,  332. 
Radical  sign,  133. 
Radicals,  331-361. 

addition  of,  341. 

division  of,  344. 


Radicals,  involution  and  evolution  of, 
355. 

multiplication  of,  344. 

order  of,  332. 

reduction  of,  333. 

similar,  341. 

simplest  form  of,  337. 

subtraction  of,  341. 
Radicand,  332. 

Ratio,  in  geometrical  progression,  434. 
Ratio  and  proportion,  218-236. 
Rational  numbers,  321. 
Rational  polynomial,  161,  162. 
Rational  term,  161. 
Rationalizing  denominator,  352. 
Rationalizing  factors,  349,  350. 
Real  numbers,  331. 
Reciprocal,  190. 
Rectangle,  1,  124. 
Reduction,  of  fractions,  168. 

of  radicals,  333. 
Remainder  theorem,  494. 
Removal  of  parenthesis,  59. 
Repeating  decimals,  437. 
Root,  cube,  5. 

extraneous,  363. 

of  equation ,  8,  156. 

of  radicals,  355. 

square,  5,  133,  279. 
Roots  and   coefficients    of  quadratic 

equation,  384. 
Roots,  nature  of,  385. 
Rules  and  formulas,  123. 

Series,  defined,  427. 

infinite  geometrical,  437. 

sum  of  terms  of,  428,  435. 
Sign,  radical,  133. 
Signed  numbers,  addition  of,  18,  20. 

defined,  18. 

division  of,  27. 

multiplication  of,  25. 

subtraction  of,  22. 
Signs,  in  division,  27,  93. 

in  fractions,  171. 

in  multiplication,  26,  69. 
Similar  radicals,  311. 
Similar  terms,  35. 
Simple  equations,  104-131. 

defined,  107. 


Index 


503 


Simultaneous  equations,  250-278,  305- 
314. 

denned,  251. 
Solution  of  problems,  116. 
Square,  completing  the,  294. 
Square  root,  5,  133,  279-289,  35(3-361. 
Substitution,  elimination  by,  259. 
Subtraction,  50-66. 

defined,  16,  22. 

of  fractions,  179. 

of  imaginary  numbers,  369. 

of  like  monomials,  50. 

of  polynomials,  54. 

of  radicals,  341. 

of  signed  numbers,  22. 

of  unlike  monomials,  52. 
Subtrahend,  22,  23. 
Surd,  conjugate  quadratic,  350. 

defined,  332. 

quadratic,  350. 
Symbols,  of  deduction,  5. 

representing  numbers,  1. 

representing  operations,  2. 
Symmetrical  equations,  407. 
Synthetic  division,  496. 

Table  of  logarithms,  461-463. 
Term,  absolute,  290. 

defined,  32. 

integral,  161. 

last,  in  progressions,  428,  434. 

rational,  161. 


Terms,  collecting,  41. 

dissimilar,  35. 

of  a  fraction,  167. 

of  a  proportion,  219. 

similar,  35. 

sum  of,  in  progressions,  428,  434. 

transposing,  110. 
Third  proportional,  220. 
Transposing  terms,  110. 
Triangle,  area  of,  1. 
Trinomial,  32. 
Type  forms  in  multiplication,  78-87. 

Unknown  number,  7. 

Unlike  monomials,  addition  of,  40, 

subtraction  of,  52. 
Unlike  terms,  35. 
Use  of  formulas,  124. 

Value,  absolute,  20. 
Variable,  448. 
Variation,  448-453. 

constant  of,  448. 

direct,  448. 

inverse,  451. 

joint,  452. 

notation,  449. 
Vinculum,  58. 

Writing  algebraic  numbers,  9. 

Zero  exponent,  318. 


LOAN  DEPT. 


L(C?795sl0)476B 


General  Library    . 


-YB   17276 
YB   17277 


M306039 


QA 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


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